1. Cake: Preheat oven to 375 degrees. Butter three 8-inch round cake pans. Line bottoms with parchment; butter parchment. Flour pans and tap out excess. Whisk together flour, baking powder, and salt in a bowl.

2. In a food processor, puree chestnuts with rum and 2 tablespoons water. Add sugars; process to combine. Add butter and vanilla; process to combine. Add egg yolks; process to combine. Transfer chestnut mixture to a large bowl; whisk in flour mixture. Whip egg whites until soft peaks form. Fold into batter.

3. Divide batter among pans; smooth tops. Bake until a tester inserted in centers comes out clean, 20 to 25 minutes. Let cool in pans on a wire rack, 10 minutes, then remove from pans and let cool completely.

4. Frosting: With a mixer on medium-high speed, beat butter until pale and creamy, about 2 minutes. Add confectioners' sugar, 1/2 cup at a time, beating well after each addition and scraping down sides of bowl as needed. Beat in vanilla and salt. Divide mixture into 2 bowls.

5. Puree chestnuts and rum in food processor until almost smooth, then stir into half of frosting. (If frosting looks slightly broken, chill, then whisk to combine.) Stir melted chocolate and cocoa into other half of frosting.

6. Place one cake layer on a cake turntable or stand. Frost cake as shown, right. Refrigerate at least 30 minutes and up to 3 days. Bring to room temperature before garnishing as desired and serving.