For the Cake: Preheat oven to 375°. Line a 16 1⁄2" × 12" heavy baking pan with buttered parchment paper, cut large enough to hang over sides of the pan by about 1". Put bittersweet chocolate in a large mixing bowl and set aside. Bring cream just to a boil in a saucepan over medium heat, then pour over bittersweet chocolate and whisk until smooth. Set aside to cool. Beat egg whites in the bowl of a standing mixer fitted with a whisk on medium speed until frothy; increase speed to medium-high and gradually add 2 tbsp. sugar, beating constantly, then increase speed to high and beat until stiff, glossy peaks form, 30-40 seconds more. (Don't over-beat.) Mix one-third of the whites into chocolate using a rubber spatula, then gently fold in remaining whites in two batches, taking care not to deflate batter. Spread in prepared pan and bake until a toothpick inserted in center comes out clean, 10–12 minutes. Set aside to cool in the pan.
For the Filling: Melt semisweet chocolate with 2 tbsp. water in the top of a double boiler set over simmering water over medium heat. Stir to combine, then set aside to cool. Combine 6 tbsp. sugar and 3 tbsp. water in a small heavy saucepan; cover and bring to a boil over medium heat, swirling pan several times until sugar has dissolved, about 1 minute. Uncover and continue to boil until syrup reaches the softball stage or 236° on a candy thermometer, about 5 minutes more. Meanwhile, beat egg yolks in the bowl of a standing mixer fitted with a whisk on high speed until thick and pale yellow, about 3 minutes. Reduce speed to medium and gradually pour in hot syrup. Beat constantly until mixture cools to room temperature, about 10 minutes. Beat 12 tbsp. butter into egg mixture 1 tbsp. at a time, waiting until completely incorporated before adding more; continue beating until thick and smooth, about 5 minutes total. Stir in cooled semisweet chocolate and set aside.
For the Icing: Melt chocolate and butter in the top of a double boiler set over simmering water over medium-low heat, whisking often. Remove from heat and gradually whisk in cream. Transfer to a medium bowl and set aside at room temperature, stirring occasionally, until icing thickens, about 4 hours. (Don't refrigerate; it makes icing hard to spread.)
To assemble the bûche: Transfer Cake with parchment to a clean work surface, sprinkle with rum, then spread Filling evenly over top using a metal spatula. Grab the long edge of the parchment paper with two hands and gently roll roulade onto itself, pulling off paper as you roll. To make stumps, diagonally cut a 2" length from each end of bûche; then, to make the stumps thinner than the bûche, partially unroll each piece, trim off flap, and discard. Set stumps aside.
Using two long metal spatulas, carefully transfer bûche to a serving platter lined with strips of waxed paper. "Glue" stumps onto bûche with some of the Icing. Melt 1⁄4 cup of Icing in a saucepan over low heat, then spoon it over stumps to coat completely. Spread remaining Icing on bûche, dragging spatula along Icing to simulate tree bark. Remove waxed-paper strips. Decorate with Meringue Mushrooms, then sift confectioners' sugar over mushrooms and bûche.