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  1. Preheat the oven to 350°F. Butter a half-sheet pan (18" x 13"), line it with parchment paper, then butter the parchment.

  2. To make the cake: Sift the cake flour and cocoa powder into a large bowl. Do not skip this step! Add the sugar, baking powder, baking soda, and salt and whisk to combine.

  3. In a medium bowl, whisk together the buttermilk, oil, eggs, egg yolk, and vanilla.

  4. Briefly stir the wet ingredients into the dry to incorporate. Add the hot water and mix just until combined; don't over-mix.

  5. Pour the batter into the prepared pan and spread into the corners with an offset spatula. Bake until puffed and a toothpick inserted in the center comes out clean, about 12 to 14 minutes. Remove from the oven and cool completely.

  6. To make the filling: Melt the chocolate over a double boiler. Stir until smooth, then remove from the heat.

  7. Beat the chestnut purée, butter, sugar, and salt until smooth. Add the melted chocolate and beat until combined.

  8. To make the frosting: Gently heat the chocolate and cream together in a double boiler or a microwave set on low power. Stir until smooth; remove from the heat.

  9. To assemble the cake: Using a serrated knife, cut the cake crosswise into four equal pieces. Carefully transfer one piece to a serving tray and spread evenly with a third of the chestnut filling (about 1 scant cup).

  10. Repeat with two more layers, placing the fourth layer on top. Pour the frosting over the top of the cake and spread it to cover. Chill the cake for 30 minutes before slicing. If not serving immediately, remove the chilled cake from the fridge to rest at room temperature for 30 minutes before serving, for best flavor.

  11. Store any leftover cake, securely wrapped, at cool room temperature for several days. Freeze for longer storage.