To make the cake: Grease three 8-by-2-inch cake pans and line with parchment round Preheat oven to 350 degrees F.
In a medium bowl whisk together the flour, baking powder and salt, set aside. In a measuring cup mix together the milk and vanilla, set aside.
In a stand mixer, fitted with the paddle attachment, mix together the sugar and oil on medium-high speed for 1 minute. Add the egg whites and mix for another minute. Add the vinegar.
Add the flour to the sugar and egg mixture in 3 batches, alternating with the milk. Mix well after each addition and scrape down the bowl. Divide the batter into 3 bowls. Add the red food coloring to one of the bowls. Repeat with the blue food coloring.
Pour the batter into the prepared cake pans. Bake for 25 to 30 minutes or until a tester comes out clean. Allow the cake layers to cool completely. Remove from the pans, leaving the parchment on the bottom. Use a 4-inch round biscuit cutter to cut out the center of each cake. Carefully lift out the center piece of cake and place them into the center of the other colored cakes.
To make the cream cheese icing: In a mixer, fitted with the paddle attachment, cream the cream cheese until it is softened, about 2 minutes. Add the butter and mix until smooth. Add the confectioners' sugar 1 cup at a time, mixing well between each addition, on low speed. This will be quite a bit of sugar to incorporate into your bowl.
To decorate the cake: Place the blue/red layer onto a cake stand or serving platter. Put one cup of the icing on the cake. Smooth the icing to cover the layer all the way to the edge.
Place the white/blue layer onto of the icing and peel off the parchment, if it is stuck to the bottom of the cake. The parchment can be useful for moving the layers. Add another cup of the icing and spread it smooth. Add the final red/white layer.
Cover the entire cake with a crumb coat of icing, to keep the crumbs from getting on the outside of your cake. Use a large star shaped cookie cutter to create the decoration on the top of the cake with blue decorating sugar. Once it is covered well, carefully lift off the cookie cutter.
Divide the remaining icing into 2 bowls and color one of the bowls with red food coloring. Fill 2 pastry bags, fitted with a star tip, with the red and white icings. Decorate around the blue decorating sugar with small red stars.
Pipe a border of white stars along the edge of the cake. Then fill in the entire top with white stars, so the top is completely covered.
Starting at the bottom of the cake, pipe a row of white stars, then repeat with a layer of red, and continue all the way up the sides of the cake.