For the cake:
2 sticks (1 cup) unsalted butter, at room temperature, plus more for greasing pans
1½ cups all-purpose flour, plus more for dusting pans
1½ cups cake flour
1 tbsp. baking powder
½ tsp. salt
1¾ cups sugar
4 large eggs
2 tsp. vanilla extract
1¼ cups milk
For the frosting:
1 lb. semisweet chocolate, finely chopped
6 tbsp. Dutch-process cocoa powder
6 tbsp. boiling water
3 sticks (1½ cups) unsalted butter, at room temperature
½ cup confectioners’ sugar, sifted
Pinch of salt
To make the cakes, preheat the oven to 350˚ F. Line the bottoms of two 9-inch round cake pans with parchment paper. Butter and flour the edges of the pans, tapping out the excess; set aside. In a medium bowl, combine the all-purpose flour, cake flour, baking powder, and salt; whisk together to blend well and set aside.
In the bowl of an electric mixer, combine the butter and sugar. Beat on medium-high speed until light and fluffy, about 4 minutes, scraping down the bowl as needed. Mix in the eggs one at a time, beating well after each addition. Blend in the vanilla. With the mixer on low speed, add in the dry ingredients in three additions, alternating with the milk. Beat each addition just until incorporated.
Divide the batter between the prepared baking pans. Bake, rotating the pans halfway through baking, until the cakes are golden and a toothpick inserted in the center comes out clean, 30-35 minutes. Transfer the pans to a wire rack and let cool 20 minutes. Run a knife around the edge of the pans to help remove the cakes. Invert the cakes onto the rack and peel off the parchment. Let the cakes cool completely before frosting. Level the cakes if necessary.
To make the frosting, place the chocolate in a heatproof bowl set over a pan of simmering water. Heat, stirring occasionally, until the chocolate has melted. Remove from the heat and let cool to room temperature, about 25-30 minutes. Meanwhile, combine the cocoa powder and boiling water in a small bowl; stir until the cocoa is dissolved.
In the bowl of an electric mixer, beat the butter, confectioners’ sugar and salt on medium-high speed until light and fluffy, 3-4 minutes. Add the melted chocolate; beat on low speed until combined, 1-2 minutes. Beat in the cocoa mixture until well blended. (Note: My frosting was still a bit runny for decorating at this point. I let it cool longer before frosting the cake so it could firm up slightly.)
To assemble the cake, place one of the cake layers on a serving platter. Top with 1 cup of the chocolate frosting and smooth over the top of the cake. Top with the remaining cake layer. Frost the top and sides of the cake with the remaining frosting as desired. Refrigerate, covered, for up to 3 days. Let come to room temperature at least 30 minutes before serving.