Step 1Heat the oven to 350 degrees. Generously grease the interior of two (9-inch) round cake pans, bottom and sides, with butter. If possible, line the bottom with parchment paper and grease the paper as well.
Step 2Sprinkle about 2 teaspoons flour into each pan and shake around until the interior is thoroughly dusted. Turn the pans upside down over the sink and tap against the sink to dislodge any excess flour.
Step 3In a small bowl, stir the cocoa into the boiling water, stirring until smooth. Set aside until it cools to room temperature.
Step 4Meanwhile, in the bowl of a stand mixer, or in a large bowl using an electric mixer, beat the 1 cup butter until fluffy, 1 to 2 minutes.
Step 5Reduce the speed and add the sugar with the mixer running, continuing to mix until light and fluffy, 2 to 3 minutes more. Add the eggs one at a time and beat into the mixture, then add the vanilla and mix until combined.
Step 6In a large bowl, combine the remaining flour with the baking powder and salt, stirring with a whisk or fork until combined.
Step 7Add one-third of the cocoa to the butter mixture and beat until combined, then one-third of the flour mixture. Repeat with the remaining cocoa and flour in two batches.
Step 8Divide the batter between the two prepared cake pans, smooth the surfaces with a spatula and bake in the center of the oven until the surface springs back when touched and the cake has started to pull away from the walls of the pan, about 30 minutes.
Step 9Transfer the cake pans to cooling racks or folded towels for 10 minutes, then overturn the pans onto a rack or plate and tap lightly to remove the cakes from the pans and paper. Turn the cakes right side up again and cool to room temperature.
Step 1In the bowl of a stand mixer, or in a large bowl using a hand mixer, beat the butter until light and fluffy, 1 to 2 minutes over low speed. Add confectioner's sugar and continue to beat until evenly incorporated. With mixer running, add salt and peppermint extract and mix. Add one-fourth cup of the milk and beat into the frosting, adding additional milk 1 tablespoon at a time, until the frosting has a light, fluffy, spreadable consistency. This makes a scant 8 cups frosting.
Step 2Assemble the cake: Put 1 cake layer, top side down, on serving plate. Cover the exposed surface with about 2 cups frosting, and spread the frosting evenly over the top of the cake to make the filling. Place the second cake layer on top of the frosting and frost the sides and top with the remaining frosting.