Special equipment: a pastry bag fitted with the No.234 grass or hair tip
Cut and assemble the cake on a 13-by-9-inch cake board according to the diagram, using American Buttercream frosting to adhere the cake pieces together. Frost the entire cake with a thin and even coating of American Buttercream. Refrigerate so the frosting can set, about 1 hour.
Dye the remaining American Buttercream using gel food coloring in the color you would like your dog to be. Place into a pastry bag fitted with a large grass tip and pipe "fur" all over the cake. Roll the white fondant into two 2-inch discs. Roll the black fondant into two small balls, then flatten each slightly and place 1 into the center of each white disc (adhere with a dab of water). Stick to the cake where you want to place the eyes. Make a teardrop with a 2-inch round of the black fondant, pinching to create nostrils; stick to the cake. Finish with an oval shape with the pink fondant, using the back of a paring knife to create a center line in the tongue' stick to the cake board coming from the dog's mouth.
For the vanilla cake: Preheat the oven to 350 degrees F. Spray the bottom and sides of four 8-inch round cake pans with nonstick cooking spray; dust with flour, and tap out the excess. Line 3 regular muffin cups and 1 mini muffin cup with paper liners.
Whisk together the flour, baking powder and salt in a medium bowl; set aside. Beat the butter, sugar and oil in a large bowl with an electric mixer on medium-high speed until light in color and fluffy, about 5 minutes. Beat in the eggs, one at a time. Beat in the vanilla until combined.
Transfer the creamed mixture to a very large mixing bowl, then alternate folding in the flour mixture and the milk with a spatula, adding the flour in 3 additions and the milk in 2, starting and ending with the flour, just until smooth. Fill the 3 muffin cups and 1 mini muffin cup halfway with batter. Divide the remaining batter evenly among the prepared cake pans.
Bake, rotating the cake pans halfway through, until a toothpick comes out clean, 8 minutes for the mini cupcake, 11 minutes for the regular, and 30 to 35 minutes for the cakes. Cool the cakes in the pans on a rack for 10 minutes, then remove the cakes to the rack to cool completely.
For the chocolate cake: Preheat the oven to 350 degrees F. Spray four 8-inch cake pans with nonstick cooking spray and dust with flour, then line the bottoms with parchment. Line 3 regular muffin cups and 1 mini muffin cup with paper liners.
Whisk together the flour, cocoa, baking soda and salt together in a large bowl.
Beat together the butter and sugar until light and fluffy, 1 to 2 minutes. Add the eggs, one at time, and beat until incorporated. Add the sour cream and continue to beat until fully incorporated. Add the vanilla. Transfer the mixture to a large mixing bowl.
Stir together the hot water and instant espresso powder in a spouted measuring cup. Whisk the dry ingredients by hand into the wet ingredients in 2 additions, alternating with the espresso and mixing just until fully incorporated.
Fill the 3 muffin cups and 1 mini muffin cup halfway with batter. Divide the remaining batter evenly among the 4 prepared cake pans. Bake until a toothpick comes out clean, 8 minutes for the mini cupcake, 11 minutes for the regular cupcakes and 30 to 35 minutes for the cakes (rotate the pans halfway through). Cool the cakes in the pans on a rack for 10 minutes, then remove the cakes to the rack to cool completely.
In the bowl of a stand mixer fitted with the whisk attachment, combine the butter, powdered sugar, vanilla, salt and milk. Whip the mixture on low speed for a while. When the sugar is incorporated, turn the mixer speed up and whip until the buttercream is smooth. Add milk until desired consistency is reached.