Special equipment: four 7-inch cake pans
Position 2 oven racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line four 7-inch cake pans with parchment rounds and spray them with nonstick spray; set aside.
Mix together the flour, granulated sugar, cocoa, baking powder, baking soda and salt in a large bowl or the bowl of a stand mixer. Stir in the buttermilk and vegetable oil. Dissolve the espresso powder in 1 cup boiling water, then stir into the mixture. Add the eggs, vanilla and vinegar and mix together on medium speed until well combined. Distribute the cake batter evenly among the prepared baking pans and bake, switching and rotating the positions of the cake pans halfway through, until a cake tester inserted in the centers comes out clean, 25 to 30 minutes. Remove from the oven and allow to cool for 10 minutes, then remove from the pans and cool completely.
Place 1 cake layer on a cake plate and frost with the Cookie Dough Buttercream; top with another layer and continue in this manner until you have used all the layers. Frost the outside with buttercream and serve.
Beat the butter and brown sugar together with a mixer on medium-high speed until fluffy, 2 minutes. Reduce the speed to low and gradually add the confectioners' sugar 1 cup at a time. Then mix in the vanilla and salt. Beat in the cream to loosen the frosting.