Preheat the oven to 350°F (175°C) with racks in middle and lower third positions.

Combine devil's food cake mix, 1 cup milk, 1/3 cup melted butter, 3 eggs, and 1 teaspoon vanilla extract. Beat at low speed for 1 minute, then medium for 2 minutes.

Spoon batter evenly into 2 nonstick 9-inch round cake pans greased with baking spray.

Repeat batter preparation with the white cake mix, substituting 3 egg whites for whole eggs.

Bake all 4 cake layers for 20-25 minutes, rotating pans halfway through. Cool for 10 minutes before removing from pans. Let cakes cool completely (about 1 hour).

Trim the tops of cooled cakes to level them.

Beat softened butter for frosting until creamy. Add cream cheese and beat until smooth. Gradually incorporate powdered sugar and cocoa powder with milk and vanilla until desired consistency is achieved.

Cut and layer cakes to assemble a checkerboard pattern, alternating chocolate and vanilla layers as described.