Martha's modern version of the traditional yule log cake, or buche de noel, calls for rolling and filling chiffon cake with chocolate-hazelnut cream, covering in chocolate "tree bark," and decorating with candied "mushrooms" and pistachio "moss."
Chiffon Cake: Preheat oven to 350 degrees. Brush two rimmed baking sheets with butter. Line with parchment; butter parchment.
In a large bowl, whisk together flour, 3/4 cup granulated sugar, baking powder, and salt. In another large bowl, whisk together oil, egg yolks, milk, and vanilla seeds. Whisk egg-yolk mixture into flour mixture.
In the bowl of an electric mixer fitted with the whisk attachment, beat egg whites on high speed until frothy. Add cream of tartar and beat until soft peaks form. Gradually add remaining 3/4 cup granulated sugar, beating until stiff, glossy peaks form, about 5 minutes. Whisk one-third of egg-white mixture into batter. Gently but thoroughly fold in remaining egg-white mixture with a rubber spatula.
Divide batter evenly between prepared sheets and spread to edges with a spatula. Bake until tops spring back when lightly touched, about 18 minutes. Dust two kitchen towels with confectioners' sugar. Invert one cake onto a towel; remove parchment. Dust with more confectioners' sugar. Starting at one short side, roll into a log, incorporating towel. Repeat with remaining cake. Let cool completely.
White Modeling Chocolate: Stir together white chocolate and 3 tablespoons corn syrup until mixture has consistency of thick batter or very soft dough. If mixture is too thick, add remaining 1 tablespoon corn syrup. Pour mixture into center of a large piece of plastic wrap; use edges of plastic to gather into a disk. Wrap tightly and let stand until firm, about 1 hour or up to overnight. When ready to use, transfer to a surface dusted with confectioners' sugar; knead until smooth and pliable.
Semisweet Modeling Chocolate: Stir together semisweet chocolate and corn syrup until mixture has consistency of thick batter or very soft dough. Pour mixture into center of a large piece of plastic wrap; use edges of plastic to gather into a disk. Wrap tightly and let stand until firm, about 1 hour or up to overnight. When ready to use, transfer to a surface dusted with cocoa; knead until smooth and pliable.
Chocolate-Hazelnut Filling: In a small bowl, whisk together chocolate-hazelnut spread and 1/4 cup cream until thickened. In a large bowl, whisk together remaining 3 cups cream and confectioners' sugar until soft peaks form. Add chocolate mixture and continue whisking until stiff peaks form.
Assembly: Unroll cooled cakes, remove kitchen towels, and reroll. Cut each cake crosswise into thirds. Unroll one piece of cake and spread with about 1 cup filling. Roll into a tight spiral and place, cut-side up, in center of a large serving platter. Unroll a second piece of cake, spread with another 1 cup filling, and wrap around perimeter of first piece. Repeat with remaining 4 pieces of cake. Spread remaining filling over top and sides of cake.
Roll white modeling chocolate into a 1/4-inch-thick round, about 11 inches in diameter. Place on top of cake and fold overhanging edges down sides of cake. Lightly brush overhanging edges with water. Roll 1 1/2 cups semisweet modeling chocolate into a 6-by-8-inch rectangle. Use the handle of a wooden spoon to press a pattern crosswise on semisweet chocolate to resemble bark. Use a pizza wheel to cut a straight edge along one long side of semisweet chocolate. Affix bark to perimeter of cake, cut-side up, and gather bottom to resemble roots, placing small balls of semisweet chocolate in between bark and cake to create more dimension. Continue in this manner, working around sides of cake. Repeat process with remaining semisweet chocolate, then seam pieces of semisweet chocolate together by pinching and folding over each other to create a gnarled appearance.
Using a paring knife, score lines onto surface of tree stump, radiating out from center, to resemble cracks. Use a soft bristled brush to dust white chocolate with cocoa. Mix a small amount of cocoa with water, one drop at a time, until it has consistency of thick paint. Dip the tip of a 2-inch-wide coarse brush into cocoa mixture; dry-brush tree rings onto surface of tree stump. Brush sides and roots of stump in some spots with corn syrup; sprinkle pistachios over brushed spots to create appearance of patches of moss. Place a few candy mushrooms on sides of cake. Decorated cake can be stored in refrigerator, covered, up to 1 day.