Preheat oven to 350°F. Butter and flour 8-inch round pans and line with parchment paper. Since you'll need 5 layers total, use as many 8-inch pans as you have and bake in batches if necessary.
In a medium bowl, combine the cake flour, salt and baking powder. In a separate bowl combine the milk, vanilla and egg whites. Set both aside.
In the bowl of a stand mixer fitted with a paddle attachment, beat the butter and sugar on medium high speed until light and fluffy, about 3-4 minutes. Scrape bowl down with a spatula a few times as needed. Add the dry ingredients in 3 additions, alternating with the wet ingredients, and ending with the dry. Mix only until incorporated.
Divide batter between 5 small bowls. For even layers, weigh the batter and divide evenly. Leave one bowl of batter as is and using the gel food coloring, tint the remaining bowls as such: 1) bright red, 2) light red, 3) bright blue, 4) light blue.
Pour batters into individual prepared cake pans and bake until centers spring back when touched, about 14-16 minutes. Bake in batches as needed to finish all 5 layers. Let cake layers cool in the pan 5 minutes, then transfer to a wire rack to cool completely.