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To Make the Soaking Syrup: Combine sugar and water in a small saucepan or in microwave safe measuring cup. Either way, stir and heat until hot enough so the sugar dissolves. Stir in whiskey; set aside.
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To Make the Chestnut Purée: Combine the chestnuts, water and sugar in a medium-sized pot and bring to a boil over medium-high heat. Adjust heat down and simmer for 25 to 35 minutes until the majority of the liquid is evaporated. Remove from heat and add the vanilla.
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Strain the chestnuts, reserving the sugar syrup, and transfer chestnuts to a food processor fitted with a metal blade. Pulse on and off and then blend until beginning to smooth out. Add reserved syrup slowly until purée is thick but smooth and spreadable, pulsing on and off until combined. Cool before refrigerating.
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Purée may be refrigerated in an airtight container for up to 3 days. If using immediately, cool before incorporating into buttercream. If refrigerated, bring to room temperature before beating into buttercream (see below).
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For the Cake: Position rack in center of oven and preheat oven to 375°F/190°C. Line a rimmed half-baking sheet pan (approx. 18 inches x 13 inches; 46 cm x 33 cm) with parchment paper. Coat paper with nonstick spray and dust lightly with extra flour, tapping out excess.
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Whisk 3/4 cup plus 1 tablespoon (125 g) flour, baking powder and salt together in a small bowl to aerate and combine; set aside.
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Place milk and butter in a saucepan over medium heat to melt butter, or melt together in the microwave. Stir in vanilla extract; set aside, keeping warm.
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Meanwhile, beat eggs, yolks and sugar with balloon whip on high speed until light and fluffy and a ribbon forms, about 2 minutes. (You can use a hand-held mixer, beaters and large bowl but mixing times will be longer). Sift the flour mixture over the eggs, a little bit at a time, and fold in gently; use a whisk to begin the folding action - this helps retain the volume of egg mixture - then switch to a silicone spatula.
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Dribble the warmed milk/butter mixture over the batter a little at a time, folding all the while. If you add the liquid too quickly, it will sink, won’t incorporate properly and you’ll be left with a rubbery layer in your cake. Scrape batter into prepared pan, using an offset spatula to spread evenly. The cake will be thin.
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Bake for about 9 to 12 minutes, rotating front to back once during baking. Cake should be puffed, lightly golden brown, and a toothpick will test clean. Do not overbake or it will lose its flexibility. Cool on a rack for 5 minutes.
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Sift a very light layer of extra flour over a clean piece of parchment or a large kitchen tea towel. Loosen cake from pan sides with an offset spatula or knife tip. Center cake over the parchment/towel and unmold onto it. Peel off parchment. Dust cake lightly with a bit more flour. Starting with a long side, roll up cake in parchment/towel, which will be rolled up inside (see pic). Cool completely.
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For Assembly: Have buttercream soft and ready to use. Beat 2/3 cup (165 ml) of room temperature chestnut purée into 1 cup (240 ml) of buttercream for the filling. Any extra chestnut purée can be saved and spread on toast.
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Unroll your cake and use a pastry brush to pat the Soaking Syrup evenly over the cake.
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Spread the Chestnut Buttercream in an even layer leaving ¼ inch (6 mm) border on one of the long sides. Roll the cake and buttercream up in a tight spiral, starting from the long end that has the buttercream all the way to the edge. Place cake seam side down on a large platter and use hands to gently squeeze into a nice cylindrical shape. Cover with plastic wrap and refrigerate until firm, about 2 hours or overnight. Roll can also be frozen at this point up to 1 day, wrapped well in plastic wrap.
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Combine remaining amount of buttercream with melted chocolate beating well until smooth and incorporated.
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Slice off an end of the log at an angle to become a “branch” along the side of the log as seen in the image. Simply press it into place; buttercream can be used as “glue”.
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Cover the log and branch with chocolate buttercream, with a small offset icing spatula; use the tines of a fork to make bark-like marks. Cake may be served immediately or let it sit at a cool room temperature for up to 6 hours. It may be refrigerated overnight, but bring to room temperature before serving.
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Place Meringue Mushrooms here and there, if you want. May also be dusted lightly with confectioners’ sugar to resemble snow. I also rolled a few cranberries in sugar and used some non-edible leaves as decoration.