Center two racks in the oven and preheat the oven to 350°F. Spray three 8 or 9 inch round cake pans with nonstick spray. Add a spoonful of flour to each, shake it around and tap out the excess. (You can also use the baking spray that already has flour in it).
Sift together the flour, baking powder and salt in a medium bowl. In another bowl, whisk together the buttermilk and egg whites.
In a large bowl, use an electric mixer to blend together the butter and sugar; beat at medium speed for a full 3 minutes, until the mixture is very light. Beat in the extract, then add one third of the flour mixture, still beating on medium speed. Beat in half of the buttermilk-egg mixture, then beat in half of the remaining dry ingredients until incorporated. Add the rest of the buttermilk and eggs, then add the last of the dry ingredients. Finally, give the batter a good 2 minute beating to ensure that it is thoroughly mixed and well aerated.
Divide the batter evenly between three glass bowls. Color two of the bowls of batter, one dark red and one dark blue. Just add the color a little at a time until you reach the hue you'd like. Scoop each of the batters into their own prepared pan.
Bake for 20 to 25 minutes for 8 inch rounds (15 to 20 for 9 inch), or until the cakes have risen and are springy to the touch. A toothpick inserted into the centers should come out clean. Transfer cakes to cooling racks and cool for about 5 minutes, then run a knife around the sides of the cakes. Invert and cool to room temperature right side up on wire racks.
When cooled, frost the cake as you would a 3 layer cake, frosting each layer before adding the next, and then covering the whole cake with frosting. For frosting recommendations see the recipe notes below.
Notes
Don't use liquid food coloring to dye the cake batter. You'll end up adding too much liquid to the mixture and it won't turn out great. Gel paste coloring can be found at baking supply stores or craft stores. It's concentrated, and it will give you the deep red and blue hues you're looking for in this cake.