Birch De Noël Recipes

BUCHE DE NOEL

Steps:

• To make the buttercream: Whisk the egg whites and sugar together in the bowl of an electric mixer. Set the bowl over simmering water and whisk gently until the sugar is dissolved and the egg whites are hot. Attach the bowl to the mixer and whip with the whisk on medium speed until cooled. Switch to the paddle and beat in the softened butter and continue beating until the buttercream is smooth. Dissolve the instant coffee in the liquor and beat into the buttercream.
• Turn the genoise layer over and peel away the paper. Invert onto a fresh piece of paper. Spread the layer with half the buttercream. Use the paper to help you roll the cake into a tight cylinder Transfer to baking sheet and refrigerate for at least 30 minutes or until set. Reserve the remaining buttercream for the outside of the buche.
• To make the marzipan: Combine the almond paste and 1 cup of the sugar in the bowl of the electric mixer and beat with the paddle attachment on low speed until the sugar is almost absorbed. Add the remaining 1 cup sugar and mix until the mixture resembles fine crumbs. Add half the corn syrup, then continue mixing until a bit of the marzipan holds together when squeezed, adding additional corn syrup a little at a time, as necessary; the marzipan in the bowl will still appear crumbly. Transfer the marzipan to a work surface and knead until smooth.
• To make marzipan mushrooms: Roll 1/3 of the marzipan into a 6-inch long cylinder and cut into 1-inch lengths. Roll half the lengths into balls. Press the remaining cylindrical lengths (stems) into the balls (caps) to make mushrooms. Smudge with cocoa powder. To make holly leaves: Knead green color into 1/2 the remaining marzipan and roll it into a long cylinder. Flatten with the back of a spoon, then loosen it from the surface with a spatula. Cut into diamonds to make leaves, or use a cutter.
• To make holly berries: Knead red color into a tiny piece of marzipan. Roll into tiny balls.
• To make pine cones, knead cocoa powder into the remaining marzipan. Divide in half and form into 2 cone shapes. Slash the sides of cones with the points of a pair of scissors.
• Unwrap the cake. Trim the ends on the diagonal, starting the cuts about 2 inches away from each end. Position the larger cut piece on the buche about 2/3 across the top. Cover the buche with the reserved buttercream, making sure to curve around the protruding stump. Streak the buttercream with a fork or decorating comb to resemble bark. Transfer the buche to a platter and decorate with the marzipan. Sprinkle the platter and buche sparingly with confectioners' sugar "snow."
• Storage: Keep at cool room temperature. Cover leftovers loosely and keep at room temperature.
• Set rack in the middle of the oven and preheat to 400 degrees.
• Half-fill a medium saucepan with water and bring it to a boil over high heat. Lower the heat so the water is simmering.
• Whisk the eggs, yolks, salt, and sugar together in the bowl of a heavy-duty mixer. Place over the pan of simmering water and whisk gently until the mixture is just lukewarm, about 100 degrees (test with your finger). Attach the bowl to the mixer and with the whisk attachment, whip on medium-high speed until the egg mixture is cooled (touch the outside of the bowl to tell) and tripled in volume.
• While the eggs are whipping, stir together the flour, cornstarch, and cocoa.
• Sift 1/3 of the flour mixture over the beaten eggs. Use a rubber spatula to fold in the flour mixture, making sure to scrape all the way to the bottom of the bowl on every pass through the batter to prevent the flour mixture from accumulating there and making lumps. Repeat with another 1/3 of the flour mixture and finally with the remainder.
• Scrape the batter into the prepared pan and smooth the top. Bake the genoise for about 10 to 12 minutes, or until well risen, deep and firm to the touch. (Make sure the cake doesn't overbake and become too dry, or it will be hard to roll.)
• Use a small paring knife to loosen the cake from the sides of the pan. Invert the cake onto a rack and let the cake cool right side up on the paper. Remove the paper when the cake is cool.
• Storage: Wrap in plastic wrap and refrigerate for several days, or double-wrap and freeze for up to a month.

BUCHE DE NOEL

Steps:

• For the walnut biscuit:
• Preheat oven to 350 degrees F. Butter a half sheet pan, line with parchment paper, butter the paper and dust with flour. Whip the egg whites to soft peaks. Beat in both sugars and whip to a stiff, glossy meringue. Alternately fold the cake flour and egg yolks into the meringue in 3 batches, starting and ending with the flour. Fold in the nuts. Spread the batter evenly in the pan, and bake until the cake is pale gold, the center springs back when you press it lightly with your finger, and the edges start to pull from the sides of the pan, 10 to 12 minutes. ;
• For the pastry cream:
• Whisk the yolks, cornstarch, and the 40 grams sugar in a medium bowl; the mixture will be very thick, but try to whisk enough to remove most of the lumps. Put the 75 grams sugar in a medium saucepan and cook over medium heat until dark brown; don't worry if it crystallizes a bit. Turn the heat to low and whisk in the maple syrup, then whisk in the whisky, turn up the heat, and let simmer for 1 to 2 minutes to boil off the alcohol. Whisk the milk and vanilla bean scrapings into the caramel mixture and bring to simmer. Slowly whisk about half of the hot mixture into the yolks, then whisk that mixture back into the pot, bring to a boil, whisking, and cook, still whisking, until thickened. Remove from the heat and whisk in the butter, salt and maple extract. Pour into a shallow dish, press plastic wrap onto the surface, and chill until set and very cold, about 4 hours. Beat the cold pastry cream in a standing mixer until smooth. When ready to use, whip the cream until it is very stiff, then beat into the pastry cream. Chill until ready to use. ;
• For the buttercream:
• Put the sugar in a medium pot and add enough water just to moisten; use your fingers to wet the sugar evenly. Bring to a boil. While the sugar is heating, start beating the yolks and egg in a standing mixer with the whisk attachment. When the syrup reaches about 240 degrees F on a candy thermometer (softball stage), pour it into the yolks with the mixer still running, taking care not to pour it onto the whisk. Beat until cooled to room temperature. Beat in the whisky and the maple extract. Cream the butter in another mixing bowl using the paddle attachment. Beat in the cooled egg mixture until smooth. You can use it right away, or chill it overnight; if you chill it, rebeat when you are ready to assemble the cake. ;
• For the sugared cranberries:
• Have a cookie sheet or shallow dish and a slotted spoon next to the stove. Put the sugar in a medium pot and add enough water just to moisten; use your fingers to wet the sugar evenly. Add the cinnamon stick. Bring to a boil, add the cranberries, and immediately pull from the heat. Transfer the cranberries to the cookie sheet with the slotted spoon. Cool. ;
• For the maple tuiles:
• Cream the butter, maple syrup, vanilla bean scrapings and salt until smooth. Scrape the sides of the bowl and beat in egg whites until smooth. Beat in the flour. Let the batter rest and hour or so at room temperature, or overnight in the refrigerator. Bring to room temperature before baking. Preheat the oven to 325 degrees F. Line a sheet pan with a silpat. Using a stencil, smear the batter thinly on the silpat and bake until dark golden brown, about 6 to 8 minutes. Remove from the pan while still warm. If you want to shape the cookies, put them over a bottle or rolling pin while warm; let cool. If desired, brush lightly with luster dust. ;
• Assembly:
• Turn the cooled cake onto a piece of parchment or waxed paper with a long side near you. Spread the chilled pastry cream evenly over the surface, leaving a 2-inch border across from you. Using the parchment paper to lift, roll the cake as tightly as possible. Set seam side down on a platter or large pan, and chill for an hour or so. Frost the cake with the buttercream, smoothing the surface so it looks like bark. Decorate with candied cranberries, walnuts, orange peel, and tuiles. ;

BIRCH DE NOëL

Steps:

• Make the cake: Preheat the oven to 375˚. Lightly coat an 11-by-17-inch rimmed baking sheet with cooking spray, then line with parchment paper. Beat the egg yolks, brown sugar and vanilla and almond extracts in a medium bowl with a mixer on medium-high speed until thick and pale yellow, about 3 minutes.
• Using clean beaters, beat the egg whites and salt in a large bowl with a mixer on medium speed until soft peaks form, about 1 minute. Gradually beat in the confectioners' sugar, then increase the speed to medium high and beat until stiff peaks form, 2 to 3 minutes. Fold one-quarter of the beaten egg whites into the egg yolk mixture with a rubber spatula. Gently fold the egg yolk mixture into the remaining egg whites until there are no more white streaks.
• Whisk the two flours in a medium bowl and gently fold into the egg mixture until combined. Spread the batter in the prepared pan. Bake until a toothpick inserted into the center of the cake comes out clean, 13 to 15 minutes.
• As soon as the cake comes out of the oven, dust with confectioners' sugar. Loosen the edges with a paring knife. Place another baking sheet on top of the cake and invert the cake onto it. Remove the top baking sheet and parchment. Dust more confectioners' sugar on the exposed cake. Cover with a clean kitchen towel (not terry cloth) and place the baking sheet on top. Invert the cake again so it is on the towel and remove the top baking sheet. Starting at a long end, roll up the cake and the towel together to form a log. Place seam-side down on a rack and let cool, about 45 minutes.
• Meanwhile, make the frosting: Beat the heavy cream, confectioners' sugar and salt in a large bowl with a mixer on medium-high speed until soft peaks form. Reduce the speed to medium and beat in the mascarpone and vanilla until thickened, about 30 seconds (do not overbeat). Refrigerate the frosting until the cake has cooled.
• Gently unroll the cake with a long side facing you. Spread 1 1/2 cups frosting over the cake, leaving a 1/4-inch border on the two short sides and a 1-inch border on the far long side. Dollop the cherry jam on top and gently spread to cover the frosting. Reroll the cake (without the towel) and place seam-side down on a platter. Trim both ends, then cut off one 3/4-inch-thick slice. Unroll the slice slightly to cut off 3 inches of the spiral; reroll the slice into a spiral (this will be the knot of the log).
• Frost the cake with all but 1/2 cup of the frosting, using a small offset spatula to create a bark-like texture. Place the small cake slice on the side to look like a tree knot; cover with the remaining frosting.
• Put the chocolate in a small microwave-safe bowl and microwave in 30-second intervals, stirring, until smooth. Let cool, then transfer to a small resealable plastic bag and snip a corner. Pipe thin lines of chocolate onto the cake, pressing it into the frosting with the tip of the bag. Refrigerate the cake until ready to serve.

BûCHE DE NOëL

Steps:

• Mushrooms and buttercream:
• Preheat oven to 225°F. Whisk egg whites, sugar, cream of tartar, and salt in a large heatproof bowl set over a medium saucepan of simmering water (bowl should not touch water) until egg whites are very liquid (mixture will be warm to the touch) and sugar is dissolved (rub between your fingers to check), about 5 minutes. Remove bowl from heat and beat with an electric mixer until stiff, glossy peaks form (bowl will feel cool), about 5 minutes.
• Line a rimmed baking sheet with parchment paper and dust very lightly with cocoa powder. Scoop out 1 cup meringue. Dollop heaping teaspoonfuls of meringue onto prepared baking sheet, spacing at least 1 1/2" apart.
• Lightly dust meringue with more cocoa powder and place a sheet of parchment over top. Use an offset spatula or your hands to flatten meringue into irregularly shaped disks about 1/8" thick. Bake until lightly browned and parchment peels away easily, 75-90 minutes (meringue will be soft when it first comes out of the oven but will dry and crisp as it cools).
• Meanwhile, with mixer on medium speed, beat butter into remaining meringue, adding a piece at a time and waiting until incorporated before adding more. Beat until mixture is very smooth. (If it looks very loose or curdled, just keep beating; buttercream will come back together.) Add vanilla extract and scrape in vanilla seeds; beat to combine.
• Do Ahead
• Mushrooms and buttercream can be made 2 days ahead. Store mushrooms airtight at room temperature. Cover and chill buttercream; bring to room temperature, then beat until smooth before using.
• Sponge Cake:
• Increase oven to 400°F. Coat a 26x18" rimmed baking sheet with nonstick spray and line with parchment paper, leaving overhang on long sides. Spray parchment. Whisk flour, cornstarch, and ⅓ cup cocoa powder in a small bowl.
• Bring milk, butter, oil, vanilla, and salt to a simmer in a small saucepan over medium-low heat. Keep warm over low heat.
• Meanwhile, beat eggs and egg yolks with an electric mixer on medium speed until smooth. Increase speed to high; beat until doubled in volume. With motor running, gradually add sugar; beat until very light and fluffy and mixture falls back on itself in a slowly dissolving ribbon (it should be at least quadrupled in volume), about 5 minutes.
• Reduce speed to medium and gradually stream in milk mixture. Sift one-third of dry ingredients over top; gently fold in until only a few streaks remain. Working in 2 additions, repeat with remaining dry ingredients, scraping bottom of bowl and using as few strokes as possible to keep eggs from deflating (a few streaks are fine). Scrape batter into prepared baking sheet and gently spread to edges of pan. Tap sheet lightly on counter to pop any large air bubbles.
• Bake cake until surface is puffed and springy to the touch, 10-12 minutes.
• Let cake cool in pan 2 minutes, then run a knife along short edges to loosen. Invert onto a wire rack and carefully peel away parchment. Using a fine-mesh sieve, dust cake with cocoa powder. Cover with a large kitchen towel. Place another wire rack on top and flip cake over so towel side is underneath. Remove top rack; dust exposed side with cocoa powder. Starting at one of the long sides, gently roll up warm cake inside towel. Let cake cool, seam side down, 30-35 minutes.
• Do Ahead
• Cake can be baked 1 day ahead. Store tightly wrapped in plastic at room temperature.
• Assembly:
• Carefully unroll towel and cake on a flat surface (cake will curl at the ends and may have a few small cracks but should stay in 1 piece). Position cake so the end that was in innermost part of spiral is closest to you; brush off any excess cocoa.
• Using a pastry brush, gently dab cocoa syrup over entire surface of cake; you may not use it all. Dollop bittersweet ganache over top and smooth with an offset spatula to create an even layer.
• Dollop mascarpone filling over ganache; carefully spread over surface, trying not to blend with ganache and avoiding last inch of cake along long side farthest from you.
• Using towel to lift edge nearest you, reroll cake, keeping towel on exterior. Chill, seam side down, until filling is set, about 30 minutes.
• Transfer filled cake to a baking sheet. Set aside 1/2 cup buttercream for attaching branches. Evenly spread remaining buttercream over cake with an offset spatula.
• Using a long serrated knife, trim 1/2" of cake from each end to create clean edges; discard (or eat!). Slice off a 4" piece of cake. Starting 1" from end, divide 4" piece in half, cutting at a 45° angle, leaving 1" at opposite end. Transfer log to a platter.
• Place angled side of each small piece of cake against roll to create branches, positioning one on top and the other on the side using a large dab of buttercream to secure. Cover any exposed cake on sides with more buttercream but leave cut ends exposed.
• Use spatula to create textured lines in buttercream to look like birch bark.
• Knead together marzipan and cocoa powder on a surface until smooth. Roll out on a sheet of parchment paper to less than 1/8" thick, then cut out wavy strips to look like tree knots; drape over buttercream and press gently to adhere.
• Using either a pastry bag fitted with a very small round tip or a disposable plastic bag with one corner snipped off, drizzle melted chocolate over log to mimic birch bark striations. Chill Bûche de Noël, uncovered, until ready to serve.
• Just before serving, press mushrooms perpendicularly into log in groups of 2 or 3.
• Do Ahead
• Bûche de Noël (without meringue mushrooms) can be assembled 1 day ahead. Wrap in plastic and chill. Attach mushrooms just before serving.

BUCHE DE NOEL

Steps:

• Make chocolate genoise and mousse, ganache icing, and meringue mushrooms.
• To assemble cake, carefully unroll genoise on the back side of a baking sheet (discard the plastic wrap and waxed paper, but keep the towel). Spread chocolate mousse evenly on cake to within 1 to 2 inches of one long end. Reroll cake, starting from other long end, using towel to help roll it. Cover with plastic wrap; chill until firm, about 1 hour.
• Place cake, seam side down, on a serving platter; tuck parchment around it to keep platter clean while decorating.
• Whip ganache at medium speed until it has the consistency of soft butter. Cut two wedges off ends of cake at a 45-degree angle; set aside. Ice log with a thin layer of ganache. Attach wedges on diagonally opposite sides of log. Spread ganache all over log, using a small spatula to form barklike ridges. Chill until ganache is firm, about 30 minutes.
• In the top of a double boiler or in a heatproof bowl set over simmering water, melt chocolate until smooth. Line a baking sheet with parchment. Spread melted chocolate 1/8 inch thick over parchment. Refrigerate until cold, 10 to 15 minutes. Roll paper back and forth until chocolate splinters; sprinkle over cake. Chill cake until ready to serve.
• When ready to serve, arrange meringue mushrooms around and on cake, and dust lightly with confectioners' sugar.

BIRCH DE NOEL - YULE LOG

Steps:

• Chocolate Cake:.
• Heat oven to 350°F
• Wrap coconut in a kitchen towel, and place in the oven for 15 minutes or longer.
• Remove from oven, and place on a work surface.
• Reduce oven temperature to 300°F
• Tap coconut with a hammer until it cracks in several places; break it in pieces (The white flesh should be slightly pulling away from hard shell).
• Carefully peel, keeping pieces of coconut as large as possible.
• Using a vegetable peeler peel largest pieces of coconut into long curls, and place on a baking sheet for 15 to 20 minutes, until just dry.
• Grate the smaller pieces on the box grater.
• You will need about 1/2 cup grated coconut for the cake filling.
• Heat oven to 400°F
• Line a 12x17 inch jellyroll pan with parchment paper and set aside.
• In a bowl of an electric mixer fitted with the whisk attachment, beat egg yolks on high speed until pale yellow, 4-5 minutes.
• Transfer to bowl, and set aside.
• Wash and dry mixer bowl.
• Place egg whites in bowl of electric mixer fitted with the whisk attachment on medium speed until soft peaks form, 1-2 minutes.
• Increase speed to high, and add sugar gradually, whipping until stiff peaks form.
• Transfer mixture to a large bowl.
• Using a rubber spatula, fold egg-yolk mixture into egg-white mixture; add cocoa powder and flour over top, and gently fold together.
• Pour batter into prepared pans; smooth top with an offset spatula.
• Bake until cake springs back when touched, 9-10 minutes.
• Remove from oven, and immediately turn cake out onto a wire rack lined with parchment paper.
• Peel parchment paper from top of cake, and cool completely.
• Place a clean kitchen towel on work surface.
• Transfer cake on parchment paper to towel.
• Using a pastry brush, coat cake with rum.
• Using an offset spatula spread the white-chocolate mousse over cake.
• Sprinkle with 1/2 cup freshly grated coconut.
• With a short end of cake facing you, roll carefully into a "log" shape on a kitchen towel, and secure with clothespins or large metal binder to hold the roll in a tight round shape.
• Place on a baking sheet, and refrigerate until firm, at least 2 hours.
• Remove from refrigerator, and unwrap.
• Using a serrated knife, trim the ends of the roll, cutting on the bias.
• Choose the more presentable end, and turn it side up, on the log, forming a branch or knot.
• If desired, use the other cut off end as a branch.
• Transfer the log to a serving platter.
• Slide 5-inch strips of parchment paper under the log to protect the tray while decorating.
• Coat with Seven Minute Frosting using an offset spatula.
• Remove parchment pieces.
• Decorate with Meringue Mushrooms, Snowy Chocolate Truffles, coconut shavings, and sugared Rosemary Sprigs.
• White Chocolate Mousse: Dissolve gelatin in 1/4 cup cold water, and set aside for 5 minutes.
• Place chocolate in the bowl of a food processor, and pulse until very finely chopped.
• Prepare an ice water bath; set aside.
• Place 3/4 cup cream in a small saucepan and bring just to a boil over medium-high heat.
• Add gelatin, and stir to dissolve completely.
• Pour into food processor with the motor running and process until chocolate mixture is smooth.
• Transfer to medium bowl and place over ice water bath.
• Chill until thick enough to fall from spoon in ribbons.
• Whip remaining 1 3/4 cups heavy cream to soft peaks.
• Fold into chocolate mixture (If not using immediately, refrigerate in an airtight container).
• Snowy Chocolate Truffles: Heat oven to 350°F
• Place almonds, if using, on a baking sheet, and bake until fragrant, about 10 minutes.
• Set aside to cool.
• Use a serrated knife to spread the chocolate.
• Place in a heat-proof bowl; set aside.
• In a saucepan, bring cream to a boil.
• Pour directly over chopped chocolate stirring to fully incorporate.
• Let sit for 10 minutes.
• Stir until completely melted and mixed.
• Allow to cool for 20 minutes.
• Stir in almonds, if using, and place in the refrigerator until set.
• Use a melon baller or 1 1/4 inch ice cream scoop to portion out chocolate.
• Roll the scoops of chocolate between palms to form 1-inch truffles.
• Refrigerate until ready to serve; dust with confectioners sugar before serving.
• Seven Minute Frosting: In a small heavy saucepan, combine 3/4 cup sugar, the corn syrup, and 2 tablespoons water.
• Heat over medium heat, stirring occasionally, until the sugar has dissolved.
• Rub a bit between your fingers to make sure there is no graininess.
• Raise heat to bring to a boil.
• Do not stir anymore.
• Boil, washing down sides of pan with a pastry brush dipped in cold water from time to time to prevent the sugar from crystallizing, until a candy thermometer registers 230F, about 5 minutes. (Depending on the humidity, this can take anywhere from 4-10 minutes).
• Meanwhile, in the bowl of an electric mixer fitted with the whisk attachment, whisk egg whites on medium speed until soft peaks form, about 2 1/2 minutes.
• Gradually add the remaining 2 tablespoons sugar.
• Remove the syrup from the heat when the temperature reaches 230F (it will keep rising as pan is removed from heat).
• Pour the syrup in a steady stream down the side of the bowl (to avoid splattering) containing the egg white mixture, with the mixer on medium-low speed.
• Beat the frosting on medium speed until it is cool, 5-10 minutes.
• The frosting should be thick and shiny. Use immediately.
• Sugared Rosemary Sprigs: Brush washed sprigs with a glaze of white sugar from a large egg thinned with a tablespoon of water, then coat in sugar, and let air dry.

BUCHE DE NOEL

Steps:

• Preheat oven to 375 degrees F (190 degrees C). Line a 10x15 inch jellyroll pan with parchment paper. In a large bowl, whip cream, 1/2 cup confectioners' sugar, 1/2 cup cocoa, and 1 teaspoon vanilla until thick and stiff. Refrigerate.
• In a large bowl, use an electric mixer to beat egg yolks with 1/2 cup sugar until thick and pale. Blend in 1/3 cup cocoa, 1 1/2 teaspoons vanilla, and salt. In large glass bowl, using clean beaters, whip egg whites to soft peaks. Gradually add 1/4 cup sugar, and beat until whites form stiff peaks. Immediately fold the yolk mixture into the whites. Spread the batter evenly into the prepared pan.
• Bake for 12 to 15 minutes in the preheated oven, or until the cake springs back when lightly touched. Dust a clean dishtowel with confectioners' sugar. Run a knife around the edge of the pan, and turn the warm cake out onto the towel. Remove and discard parchment paper. Starting at the short edge of the cake, roll the cake up with the towel. Cool for 30 minutes.
• Unroll the cake, and spread the filling to within 1 inch of the edge. Roll the cake up with the filling inside. Place seam side down onto a serving plate, and refrigerate until serving. Dust with confectioners' sugar before serving.

BIRCH DE NOEL

Steps:

• Preheat oven to 350°F. Test each of the three eyes at stem end of coconut to find softest one. Use an ice pick and a hammer to pierce it, then drain the juice into a glass through a fine sieve and reserve for another use. Tap coconut with hammer until a palm-size piece of shell breaks off. Continue tapping around broken edge until shell begins to break apart. Place broken shell on a baking sheet; bake about 10 min- utes to loosen flesh. Remove from oven and let cool slightly. Pry out flesh with a paring knife. With a potato peeler, remove any remaining brown skin from flesh.
• Reduce oven temperature to 250°. Using a vegetable peeler, shave largest pieces of coconut into long curls, place on baking sheet, and bake until just dry but no color has developed, 15 to 20 minutes. Meanwhile, grate smaller coconut pieces on the small holes of a box grater. (You will need about 1/2 cup grated coconut for the filling.)
• Increase oven temperature to 400°. Coat a 12 1/2-by-17 1/2-inch rimmed baking sheet with cooking spray. Line with parchment paper; coat with cooking spray. Dust with cocoa; tap out excess. Whisk together cocoa powder, flour, and salt in a bowl. In a heat-proof mixer bowl set over (not in) a pan of simmering water, whisk together eggs, yolks, and granulated sugar until sugar has dissolved and mixture is warm to the touch. Attach bowl to mixer fitted with the whisk attachment and beat on medium-high speed 2 minutes. Raise speed to high; beat until mixture is pale and thick, about 4 minutes more.
• Sift cocoa mixture over egg mixture; carefully fold in with a large rubber spatula. When almost incorporated, pour melted butter down side of bowl; gently fold to combine. Pour batter into prepared sheet, and smooth top with an offset spatula.
• Bake until cake springs back when touched, 9 to 10 minutes. Place a clean kitchen towel on a work surface; dust generously with cocoa. Remove pan from oven and immediately turn cake out onto a wire rack lined with parchment. Invert onto prepared towel; remove parchment.
• While cake is still warm, gently roll, starting from one short side, into a log, incorporating towel. Let rolled cake cool completely. (This will give it a "shape memory," so it will be easier to roll again with filling.)
• Unroll cooled cake and transfer to parchment. Using a pastry brush, coat cake with rum. Using offset spatula, spread mousse over cake. Sprinkle with 1/2 cup reserved grated coconut.
• With a short end of cake facing you, roll carefully into a roulade. Wrap in a kitchen towel; secure with clothespins or large metal binder clips to hold roll in a tight round shape. Place on a baking sheet and refrigerate until firm, at least 2 hours and up to 1 day.
• Remove from refrigerator and unwrap. Use a serrated knife to trim ends of roll, cutting on the bias (starting 1 inch in from end on shorter side, and finishing 2 inches in, to create a wedge). Choose the more presentable trimmed end and place, cut-side up, on log, forming a branch. If desired, use other trimmed end for a second branch.
• Transfer log to a serving platter. Slide strips of parchment under perimeter of log (to keep platter clean). Use offset spatula to coat log with frosting, then attach one or both trimmed ends; coat with frosting. Remove parchment pieces. Decorate with meringue mushrooms, coconut shavings, and rosemary sprigs.