Do not substitute semisweet chocolate chips for the chopped semisweet chocolate in the frosting—chocolate chips contain less cocoa butter than bar chocolate and will not melt as readily. For best results, don’t make the frosting until the cakes are cooled, and use the frosting as soon as it is ready. If the frosting gets too cold and stiff to spread easily, wrap the mixer bowl with a towel soaked in hot water and mix on low speed until the frosting appears creamy and smooth. Refrigerated leftover cake should sit at room temperature before serving until the frosting softens.
12 tablespoons unsalted butter, very soft, plus extra for greasing the pans
1 3/4 cups (8 3/4 ounces) unbleached all-purpose flour, plus extra for dusting the pans
4 ounces unsweetened chocolate, coarsely chopped
1/4 cup (3/4 ounce) Dutch-processed cocoa powder
1/2 cup hot water
1 3/4 cups (12 1/4 ounces) sugar
1 1/2 teaspoons baking soda
1 teaspoon salt
1 cup buttermilk
2 teaspoons vanilla extract
4 large eggs, plus 2 large egg yolks
16 ounces semisweet chocolate, finely chopped (see note)
8 tablespoons unsalted butter
1/3 cup sugar
2 tablespoons corn syrup
2 teaspoons vanilla extract
1/4 teaspoon salt
1 1/4 cups cold heavy cream
1. For the cake: Adjust an oven rack to the middle position and heat the oven to 350 degrees. Grease two 9-inch-round by 2-inch-high cake pans with softened butter; dust the pans with flour and knock out the excess. Combine the chocolate, cocoa powder, and hot water in a medium heatproof bowl; set the bowl over a saucepan containing 1 inch of simmering water and stir with a heatproof rubber spatula until the chocolate is melted, about 2 minutes. Add 1/2 cup of the sugar to the chocolate mixture and stir until thick and glossy, 1 to 2 minutes. Remove the bowl from the heat and set aside to cool.
2. Whisk the flour, baking soda, and salt in a medium bowl. Combine the buttermilk and vanilla in a small bowl. In the bowl of a standing mixer fitted with the whisk attachment, whisk the eggs and yolks on medium-low speed until combined, about 10 seconds. Add the remaining 11/4 cups sugar, increase the speed to high, and whisk until fluffy and lightened in color, 2 to 3 minutes. Replace the whisk with the paddle attachment. Add the cooled chocolate mixture to the egg-sugar mixture and mix on medium speed until thoroughly incorporated, 30 to 45 seconds, pausing to scrape down the sides of the bowl with a rubber spatula as needed. Add the softened butter one tablespoon at a time, mixing about 10 seconds after each addition. Add about one-third of the flour mixture followed by half of the buttermilk mixture, mixing until incorporated after each addition (about 15 seconds). Repeat using half of the remaining flour mixture and all of the remaining buttermilk mixture (the batter may appear separated). Scrape down the sides of the bowl and add the remaining flour mixture; mix at medium-low speed until the batter is thoroughly combined, about 15 seconds. Remove the bowl from the mixer and fold the batter once or twice with a rubber spatula to incorporate any remaining flour. Divide the batter evenly between the prepared cake pans; smooth the batter to the edges of the pan with a spatula.
3. Bake the cakes until a toothpick inserted into the center comes out with a few crumbs attached, 25 to 30 minutes. Cool the cakes in the pans for 15 minutes, then invert onto a wire rack. Cool the cakes to room temperature before frosting, 45 to 60 minutes.
4. To make the frosting: Melt the chocolate in a heatproof bowl set over a saucepan containing 1 inch of barely simmering water, stirring occasionally until smooth. Remove from the heat and set aside. Meanwhile, heat the butter in a small saucepan over medium-low heat until melted. Increase the heat to medium; add the sugar, corn syrup, vanilla, and salt and stir with a heatproof rubber spatula until the sugar is dissolved, 4 to 5 minutes. Add the melted chocolate, butter mixture, and cream to the clean bowl of a standing mixer and stir to thoroughly combine.
5. Place the mixer bowl over an ice bath and stir the mixture constantly with a rubber spatula until the frosting is thick and just beginning to harden against the sides of the bowl, 1 to 2 minutes (the frosting should be at 70 degrees). Place the bowl on a standing mixer fitted with the paddle attachment and beat on medium-high speed until the frosting is light and fluffy, 1 to 2 minutes. Stir with a rubber spatula until completely smooth.
6. To frost the cake: Spread a dab of frosting in the center of a cardboard round cut slightly larger than the cake. Place one cake layer on the cardboard round. Spread 11/2 cups frosting evenly across the top of the cake with a spatula. Place the second cake layer on top, then spread the remaining frosting evenly over the top and sides of the cake. Cut into slices and serve.