How to Make It
Heat oven to 350°F. Butter two 8- or 9-inch round cake pans, line the bottoms with parchment, butter again, and dust with cocoa, tapping out the excess. In a medium bowl, whisk together the flour, cocoa, baking powder, baking soda, and salt; set aside.
Using an electric mixer, beat the butter and sugar on medium-high until fluffy, 2 to 3 minutes. Beat in the vanilla, then the eggs one at a time, scraping down the sides of the bowl as necessary.
Reduce mixer speed to low. Add the flour mixture in 3 additions and the milk in 2 additions, beginning and ending with the flour mixture. Mix just until combined (do not overmix).
Transfer the batter to the prepared pans and bake until a toothpick inserted in the center comes out clean, 30 to 35 minutes for 8-inch pans and 25 to 30 minutes for 9-inch pans. Cool the cakes in the pans for 15 minutes, then turn out onto racks to cool completely.
Transfer one of the cooled cakes to a plate and spread with 1 to 1½ cups of the frosting. Top with the remaining cake and spread the top and sides with the remaining frosting. Using the palm of your hand, gently press the cookies into the sides of the cake.