Preheat the oven to 350 degrees F. For the cake: Put the eggs and 1 cup of the sugar in a large mixing bowl and, with an electric mixer fitted with a wire whisk, beat on medium-high speed until the mixture is pale yellow, thick and has tripled in volume, about 8 minutes. Sift the cocoa, flour and baking powder together in another large mixing bowl. Add the egg mixture and fold to mix thoroughly. Grease 2 (9 by 2-inch) round cake pans with the butter. Sprinkle each with a tablespoon of the remaining sugar. Pour the cake batter evenly into the pans and bake until the cake springs back when touched, about 25 minutes. Let cool for about 2 minutes. Using a thin knife, loosen the edges of the cakes and flip onto a wire rack. Let cool completely. Sift together the confectioners' sugar and cocoa powder into a medium-size bowl. Add the butter and mix with an electric mixer until incorporated. Add the vanilla and boiling water and mix until smooth. Let cool. In the bowl of an electric mixer, combine the cream and the chocolate. On medium speed. Whip until stiff peaks form, set aside.
To assemble the cake: Line a baking sheet with parchment paper and place a wire rack over it. Using a serrated knife, cut each cake in half horizontally. Brush the tops of three of the layers with 1/4 cup of the simple syrup. Place the bottom layer on a 9-inch round of cardboard and set it on the wire rack. Spread 3/4 cup of the mousse evenly on top of the cake. Top with a second layer of cake. Spread another 3/4 cup of the mousse over the top of the cake. Repeat the same process with the third layer. Top with the fourth layer. If necessary, shave off any uneven pieces of the cake with a serrated knife so that it is smooth and even on all sides. Chill for 2 hours. Spread the frosting evenly over the sides and top of the cake. Refrigerate until the frosting sets. Place the chocolate curls on top of the cake. Slice and serve with the ice cream.