Place the chocolate in a heatproof, medium bowl. In a small saucepan, bring the cream to a simmer over medium heat. Pour it over the chocolate; stir until melted and smooth. Cover; refrigerate for 1 hour.
Preheat the oven to 400 degrees . Grease a 12-by-18 inch rimmed baking sheet. Line the sheet with parchment; grease the parchment. In a large saucepan, bring 1 inch of water to a simmer over medium-high heat. In a small saucepan, melt the butter over medium heat. Add the milk and cook until warm; remove from the heat and keep warm. In a small bowl, whisk the flour, baking powder and salt.
In the bowl of a standing mixer, whisk the whole eggs, yolks and sugar by hand. Place the bowl over the pan of simmering water, making sure the bottom doesn't touch the water. Whisk until the mixture reaches 115 degrees on an instant-read thermometer, 3 to 5 minutes. Whisk in the vanilla.
Transfer the bowl to the mixer fitted with the whisk attachment. Beat at high speed until tripled in volume, about 5 minutes. Remove from the mixer. Sift one-third of the flour mixture over the egg mixture; fold in. Repeat the process twice more with the remaining flour (do not overmix). Gently fold in the milk mixture.
Transfer the batter to the prepared baking sheet, spreading evenly and into the corners. Bake until the cake is springy to the touch, about 10 minutes. Using a small, sharp knife, cut around the edges to release the cake. Place a kitchen towel on top of the cake and invert onto a cooling rack (the towel will be between the rack and the cake). Loosen the parchment, but leave it on top of the cake. Let the cake cool completely.
Using an electric mixer, beat the chilled chocolate on medium speed until thick, about 1 minute (do not overbeat). Transfer the cake on the towel to a work surface and discard the parchment. Spread with about one-third of the chocolate, leaving a 1-inch border on one long side; refrigerate the remaining chocolate.
Starting with the unfrosted end and using the towel to get started, roll up the cake jellyroll-style. Transfer the cake seam side down to a platter; refrigerate until firm, 30 minutes.
Using a serrated knife, trim a 2-inch thick piece from each end of the cake. Place on either side of the cake, creating branches. Frost the cake with the remaining chocolate mixture, leaving all 4 cut ends unfrosted. Sprinkle the nuts on top.