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To make the chocolate sponge cake: Preheat the oven to 350°F. Lightly grease the bottom (but not the sides) of a 13" x 18" baking sheet (half-sheet pan) with nonstick spray and place a piece of parchment on top. Leave the parchment ungreased.
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In a medium bowl, sift together the flour, cocoa, baking powder, and salt. Set aside.
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In the bowl of a stand mixer fitted with the whisk attachment, whisk the egg yolks, 6 tablespoons (75g) of the sugar, and vanilla on high speed until thick, pale, and the ribbon stage is achieved, about 5 to 6 minutes. Transfer the whipped mixture to another bowl and set aside.
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Clean the mixing bowl and then place the egg whites in the bowl. Whisk on medium speed until foamy, 2 to 3 minutes.
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Gradually add the remaining 6 tablespoons (75g) sugar and whip to medium-stiff peaks, 3 to 4 minutes.
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Using a flexible spatula, gently fold half of the egg white mixture (meringue) into the whipped yolks until almost completely combined. Then transfer that mixture to the stand mixer bowl with the remaining meringue, and gently fold until uniform.
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Sift the dry ingredients into the bowl in several additions; once most of the surface of the batter is covered in dry ingredients, stop sifting and gently fold them in. Repeat sifting and folding until all the dry ingredients are incorporated.
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Finish by drizzling the oil over the batter in two additions, folding it in after each, until fully incorporated. Pour the batter into the prepared pan and spread evenly with an offset spatula.
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Bake the chocolate sponge cake for 10 to 12 minutes, or until the cake springs back when lightly touched.
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Remove the cake from the oven and transfer it to a rack to cool, still in the pan. After about 5 minutes, use a thin spatula or sharp knife to release the cake from the sides of the pan. Allow the cake to continue cooling, still in the pan.
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Once cool, place a clean piece of parchment over the top of the cake. Place an upside-down baking sheet on top of the parchment and invert the cake. Remove the top baking sheet and carefully peel off the parchment. Replace it with clean parchment and place the upside-down baking sheet back over the top. Invert the cake once more and remove the top baking sheet. The cake should now be right-side up and sandwiched between two clean pieces of parchment on a baking sheet. Tightly wrap or cover the cake, still between parchment, and refrigerate until ready to assemble. The sponge can be made 1 day in advance.
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To make the chocolate bark: Temper the chocolate following the directions outlined in the Guide to Tempering Chocolate. Once the chocolate is at a workable temperature (88°F to 90°F), pour about half (225g or 3/4 cup) onto a piece of parchment. Use an offset spatula to spread the chocolate into a thin, even layer, about 1/16" to 1/8" thick; it will cover most of the parchment. Set aside the remaining chocolate for additional assembly.
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Starting from one of the long sides, roll the parchment over the melted chocolate to form a tube about 2" to 3" in diameter; as you work, make sure that only parchment touches the chocolate, no chocolate to chocolate, or it will stick together. Allow the chocolate to set in the rolled-up tube in a cool, dry place (you can slide the parchment into an empty paper towel tube to help it keep its shape), about 30 minutes. (If necessary, briefly transfer the tube to the refrigerator to set, 10 to 15 minutes.)
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Once the chocolate is set, carefully unroll the tube over a parchment-lined baking sheet and break the chocolate into desired sizes; for assembly, you’ll want some shorter 2" to 3" shards as well as some longer 4" to 6" shards. Set aside the chocolate bark until ready to assemble.
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To make the matcha moss: In a small bowl, sift together the flour, matcha powder, and baking powder. Set aside.
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In a medium bowl working with an electric hand mixer, beat the egg and sugar on medium speed until very frothy, about 1 1/2 minutes.
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Beat in the honey and salt, then continue to mix on medium for an additional 30 seconds, until the mixture is frothy.
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Using a flexible spatula, gently but quickly fold in the sifted dry ingredients.
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Divide the batter evenly between two tall microwave safe cups (16-ounce pint glasses work well here), about 1/3 cup (62g) batter into each cup.
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Microwave one cup at a time for about 45 seconds, until it has more than doubled in volume, appears spongy, and springs back when pressed.
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Remove from the microwave and immediately invert the cups onto a wire rack. Allow to cool completely while inverted (about 10 to 15 minutes), then remove the cake from the cups and store in an airtight container until ready to use, up to 1 day in advance.
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To assemble the Bûche de Noël: Fit a pastry bag with a French star tip and fill it with about 1/2 cup of the Cookie Butter Whipped Ganache. Set aside to use for decoration.
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Unwrap the prepared chocolate sponge cake, remove the top piece of parchment, and transfer the cake and the clean sheet of parchment beneath it to a work surface. Transfer the remaining Cookie Butter Whipped Ganache onto the cake and spread into an even layer.
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Starting from one of the long ends, roll up the cake, using the parchment to guide the roll. Roll tightly but gently, avoiding cracks if possible. Wrap the rolled-up cake in parchment and place it seam-side down on a baking sheet.
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Transfer the baking sheet with the cake to the refrigerator to chill for at least 30 minutes (and up to 8 hours, well-wrapped) to firm up. Alternatively, place the baking sheet with the cake in the freezer for 20 minutes to chill.
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While the cake chills, prepare the additional components: Gather the shards of tempered chocolate bark, tear the matcha moss into pieces of your desired size, and peel the pomegranate (if using arils from the whole fruit).
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Once the cake is chilled, remove it from the refrigerator or freezer, unwrap, and use a hot, dry knife to trim a thin slice off both ends of the roll to reveal a clean spiral.
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Cut two 3"-long pieces off the log, slicing on a sharp diagonal. Use the reserved whipped ganache to attach one piece to the side of the log and one piece to the top of the log to create two branches.
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Brush or pipe melted chocolate onto one of the tempered chocolate shards, then adhere it to the surface of the log, including the branch. Repeat with the remaining shards and melted chocolate until the entire log is covered. At this point, the assembled log can be covered and stored in the refrigerator overnight, if desired.
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Lightly dust the Bûche de Noël with confectioners’ sugar, then use the melted chocolate or reserved ganache to attach the mushrooms, and decorate with the moss and pomegranate seeds as desired. (Placing some of the decoration around the base of the log and some on top creates a beautiful scene.)
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For best results, serve the fully assembled cake within a couple of hours. Make clean slices by letting a sharp chef’s knife rest in a container of hot water to warm up, then dry the knife, and slice the cake slowly. (The heat from the knife makes it easy to slice through the tempered chocolate bark.) Garnish each plated slice with an extra mushroom, a patch of matcha moss, a dollop or pipette of ganache, and a few additional pomegrante seeds, if desired.