Bûche de Noël (Yule Log Cake)

For the cake:

1. Preheat oven to 350 degrees. Prepare a 10 1/2-inch by 15 1/2 inch rimmed baking sheet (jelly roll pan) by greasing with butter and lining with parchment paper, then grease and flour the parchment paper and the sides of the pan. In a medium bowl, whisk the flour, baking powder, and salt together. Set aside.
2. Using a stand mixer or a hand mixer with the paddle attachment, beat the eggs on medium speed until thick and pale, about 3 minutes. Add sugar and vanilla and continue to beat until mixture has tripled in size. Gently sprinkle the flour mixture over the eggs and fold gently until blended using a rubber spatula. Pour the batter into the prepared pan, spreading evenly. Bake until cake is done, springing back gently when lightly touched, about 13 to 15 minutes.
3. Meanwhile, on the counter, lay a clean kitchen towel down and sift powdered sugar evenly and generously over the towel. Remove the cake from the oven and loosen the cake by running a knife immediately edges of the pan. Place the cake on the prepared kitchen towel by holding the cake in place and inverting the pan onto the towel. Lift off the pan while gently peeling off the parchment paper. Roll up the cake by beginning on the long edge of the cake and roll up with the towel. Set on a wire rack to cool.

To make the syrup:

1. While the cake is baking, in a small saucepan over medium heat, combine sugar and water and stir until the sugar dissolved. Bring to a boil, then remove from heat. Stir in rum or liqueur, and set aside to allow to cool to room temperature.

To make the frosting:

1. Using a double boiler, combine the chocolate and cream in the top bowl. Set bowl with mixture over barely simmering water. Melt the chocolate and whisk mixture until blended. Remove the bowl from heat and place in refrigerator to cool, stirring occasionally, about 2 hours (see notes).
2. When the mixture has cooled, add the vanilla and salt. Mix in the bowl of a stand mixer or by hand on medium-high speed until mixture is firm enough to hold a soft dollop. As the frosting sits, it will continue to firm up.

For the meringue mushrooms:

1. Preheat oven to 225 degrees. Prepare 2 baking sheets by lining with parchment paper. Prepare a pastry bag fitted with a small number 6 plain tip.
2. In the bowl of a stand mixer or using a hand mixer on medium-low speed, beat together the egg whites and cream of tartar until foamy. Continue to beat, slowly adding granulated sugar. Increase speed to high and beat until soft peaks form and the mixture holds stiff shiny peaks. Sift powdered sugar over the egg white mixture and fold in until well blended, using a rubber spatula.
3. Scoop the mixture into the prepared pastry bag. On the first baking sheet, pipe 48 mushroom stems by piping them from about 1/2 inch at the base of the stem to about 3/4 inch at the tip, spacing them 1/2 inch apart. Pipe the tops of the mushrooms on the other baking sheet by piping 48 mounds about 1 1/4 inch wide and about 3/4 inch tall, spacing them also 1/2 inch apart. Remove any pointy tips by gently rubbing with a slightly damp finger. Dust mushrooms with cocoa.
4. Place baking sheets in the oven and bake until firm enough to be removed from sheet, about 50 to 55 minutes. Place sheets on counter (make sure to protect counter by placing on a trivet) and flip mounds so the flat sides are up. With the tip of a knife, gently make a small hole in the bottom of the mound. Pipe a small amount of reserved meringue into the hole and insert the stems tip first. Repeat with remaining mounds. Place sheets in oven and bake until completely dry, about 15 minutes. Allow to cool completely on baking sheets.

To make chocolate curls:

1. Using a vegetable peeler, shave the edge of the chocolate bar for narrow curls, or the side of the chocolate bar for wide curls. Curls can be refrigerated or kept at room temperature for a short amount of time.

To assemble the cake:

1. Carefully unroll the cake, and brush liberally with cooled prepared syrup. With an offset spatula, spread one-third of icing evenly over the cake. Gently re-roll cake (without kitchen towel) and place roll seam-side down on cutting board. Using the remaining frosting, frost the tops and sides of the roll with long, rough strokes. Trim each end of the roll with a serrated knife, cutting on a sharp angle.
2. Transfer the roll to a serving plate and garnish with chocolate curls and meringue mushrooms and dust with powdered sugar.