For the cake:
Unsalted butter for greasing
1 cup all-purpose flour, plus more for dusting
3/4 tsp. baking powder
1/4 tsp. salt
4 eggs, at room temperature
2/3 cup granulated sugar
1 1/4 tsp. vanilla extract
Confectioners sugar for dusting
For the syrup:
1/4 cup water
1/4 cup granulated sugar
2 to 3 Tbs. dark rum or coffee-flavored liqueur
For the frosting:
10 oz. bittersweet chocolate, finely chopped
2 1/4 cups heavy cream
1 tsp. vanilla extract
Pinch of salt
For the meringue mushrooms:
3 egg whites, at room temperature
1/4 tsp. cream of tartar
1/2 cup granulated sugar
1/3 cup confectioners sugar
Unsweetened cocoa powder for dusting
Chocolate curls for garnish
Confectioners sugar for garnish
Preheat an oven to 350°F. Grease a 15 1/2-by-10 1/2-inch rimmed baking sheet and line the bottom with parchment paper. Grease and flour the paper and the pan sides.
To make the cake, in a bowl, whisk together the flour, baking powder and salt until blended. In a large bowl, using an electric mixer on medium-high speed, beat the eggs until pale and thick, about 3 minutes. Add the granulated sugar and vanilla and continue beating until tripled in volume, about 3 minutes more. Sprinkle the flour mixture over the eggs and, using a rubber spatula, fold gently until just blended. Pour the batter into the prepared pan and spread evenly. Bake until the cake springs back when lightly touched, 13 to 15 minutes.
Meanwhile, lay a clean kitchen towel on the counter and sift confectioners sugar generously onto it, covering it evenly. When the cake is ready, remove it from the oven and immediately run a knife around the inside of the pan to loosen the cake. Holding the cake in place, invert the pan onto the prepared towel. Lift off the pan and carefully peel off the paper. Beginning on a long edge, roll up the cake and towel together. Set on a wire rack and let cool.
To make the syrup, in a small saucepan over medium heat, combine the water and granulated sugar and stir until the sugar dissolves. Bring to a boil and remove from the heat. Stir in the rum and set aside to cool to room temperature.
To make the frosting, combine the chocolate and cream in the top pan of a double boiler. Set the pan over but not touching barely simmering water in the bottom pan and melt the chocolate, then whisk until well blended. Remove the bowl from the heat and refrigerate, stirring occasionally, until cold, about 2 hours. To speed this process, use an ice bath (see headnote above). When the mixture is cold, add the vanilla and salt. Using an electric mixer on medium-high speed, beat the chocolate mixture briefly until firm enough to hold a soft dollop. The mixture will continue to firm up as it sits.
To make the meringue mushrooms, preheat the oven to 225°F. Line 2 baking sheets with parchment. Have ready a pastry bag fitted with a small (no. 6) plain tip.
In a bowl, using a mixer on medium-low speed, beat together the egg whites and cream of tartar until very foamy. Slowly add the granulated sugar while beating. Increase the speed to high and beat until soft peaks form when the beaters are lifted. Continue until the whites hold stiff, shiny peaks. Sift the confectioners sugar over the whites and, using a rubber spatula, fold in until well blended.
Scoop the mixture into the bag. On 1 baking sheet, pipe 48 stems, each 1/2 inch wide at the base and tapering off to a point at the top, 3D4 inch tall, and spaced about 1/2 inch apart. On the other sheet, pipe 48 mounds for the tops, each about 1 1/4 inches wide and 3D4 inch high, also spaced 1/2 inch apart. With a damp fingertip, gently smooth any pointy tips. Dust with cocoa. Reserve the remaining meringue.
Bake until dry and firm enough to lift off the paper, 50 to 55 minutes. Set the pans on the counter and turn the mounds flat side up. With the tip of a knife, carefully make a small hole in the flat side of each
mound. Pipe small dabs of the remaining meringue into the holes and insert the stems tip first. Return to the oven until completely dry, about 15 minutes more. Let cool completely on the baking sheets.
To assemble the bûche, unroll the cake and brush it liberally with the cooled syrup. Using an icing spatula, spread one-third of the frosting over the cake. Gently reroll the cake and place, seam side down, on a cutting board. Frost the top and sides of the roll with the remaining frosting, using long, rough strokes. Using a serrated knife, trim each end on a sharp angle.
Transfer the cake to a serving plate and garnish with the chocolate curls, a sifting of confectioners sugar and the meringue mushrooms. Pass the remaining meringue mushrooms at the table for garnishing individual servings. Serves 12 to 16.