MAKE THE CAKE

1. Position a rack in the center of the oven and preheat the oven to 350°F. Grease the bottom and sides of two 9-inch round cake pans. Dust the pans with flour.

2. Sift together the flour, cocoa powder, baking powder, baking soda, and salt into the bowl of an electric mixer. Add the granulated sugar and, using the paddle attachment, mix at low speed until blended.

3. In a medium bowl, whisk together the eggs, egg yolk, buttermilk, melted butter, and vanilla extract. At low speed, add the egg mixture to the dry ingredients in a steady steam. Scrape down the sides of the bowl with a rubber spatula, then beat at medium speed until well blended, about 1 minute. Add the hot coffee, mixing just until blended. Remove the bowl from the mixer stand and stir the batter up from the bottom of the bowl a few times to thoroughly blend it. Scrape the batter into the prepared pans, dividing it evenly and smoothing the tops.

4. Bake the cakes for 30 to 35 minutes, until a cake tester inserted into the center comes out clean. Cool the cakes in the pans on wire racks for 10 minutes.

5. Invert the cakes onto the racks and cool completely.

MAKE THE FILLING

6. In the bowl of an electric mixer, using the paddle attachment, beat the egg yolks, sugar, cornstarch, and salt at medium speed until pale, about 1 minute.

7. In a medium heavy saucepan, combine the water and cream and bring to a boil over medium heat. Remove the pan from the heat and whisk about half of the hot cream mixture into the yolk mixture. Whisk this mixture back into the remaining cream mixture in the saucepan. Cook over medium heat, whisking constantly, until the mixture comes to a boil. Continue to boil, whisking, for 1 minute. Remove the pan from the heat and whisk in the chocolate until completely melted.

8. Pass the filling through a fine-mesh sieve into a small bowl. Stir in the vanilla extract. Cover the surface of the pudding with plastic wrap and refrigerate for at least 2 hours, until chilled.

MAKE THE FROSTING

9. Put the chocolate in a medium stainless steel bowl and place over a pot of barely simmering water. Heat, stirring frequently, until the chocolate is completely melted. Remove the bowl from the pot and set the chocolate aside to cool until tepid.

10. In the bowl of an electric mixer, using the paddle attachment, beat the butter at medium speed until creamy, about 1 minute. Add the confectioners’ sugar and beat at medium-high speed for 2 minutes, scraping down the sides of the bowl as necessary. Beat in the vanilla extract and melted chocolate, then beat until well-blended and creamy, about 1 minute.

ASSEMBLE THE CAKE

11. Using a long serrated knife, cut each cake layer in half horizontally, to make 4 layers. Reserve one layer. Place another cake layer, cut side up, on a serving plate. Whisk the chilled filling until smooth. Using a small offset metal spatula, spread half of the filling over the layer. Top with another cake layer and spread it with the remaining filling. Top with a third cake layer, smooth side up.

12. Break half of the reserved cake layer into large pieces and place them in the bowl of a food processor (save the remainder of the cake layer for snacking). Process the cake to fine crumbs, and set aside.

13. Using a narrow metal spatula, frost the top and sides of the cake with the frosting. Pat a generous amount of cake crumbs onto the sides of the cake, pressing them lightly into the frosting. Sprinkle the remaining cake crumbs over the top of the cake. Serve at room temperature or slightly chilled.