Preheat the oven to 350°F. Line 3 (8-inch) cake pans with parchment paper, then grease and set aside.
In a medium bowl, whisk together the cake flour, salt, baking powder, and baking soda, and set aside.
In a large bowl or the bowl of a stand mixer, beat together the butter and sugar until light and fluffy. Add the eggs, egg whites, and vanilla, and mix until smooth. With the mixer on low, gradually add the dry ingredients, until well combined, scraping down the sides of the bowl as needed. Pour in the buttermilk and mix until combined.
Divide the batter into 3 bowls and add red food coloring to one and blue to another, keeping third white. Pour into three prepared cake pans. Bake for 25-30 minutes, until lightly browned around the edges, and a toothpick inserted in the center comes out clean. Remove from oven and let cool completely. To make the frosting, beat the butter in a large bowl or the bowl of a stand mixer until light and fluffy. Add the powdered sugar, milk, vanilla, and salt, and mix to combine. Add milk or powdered sugar as needed to make a smooth frosting. Beat on high for 2 minutes, until light and fluffy.
To assemble the cake, level the tops of each of the cake layers with a serrated knife. Place the blue cake layer down on a cake turntable or serving platter. Spread with one cup of the frosting. Place the white cake layer on top and spread with another cup of frosting. Place the red layer on top, then spread the frosting around the top and sides.
For a smoother look, add a light layer of frosting (known as a crumb coat) around the top and sides of the cake, then refrigerate for 20-30 minutes. Once the cake has chilled slightly, spread a thicker layer of frosting around the sides and over the top, and add dollops of frosting for decoration, if desired. Sprinkle with patriotic sprinkles. Refrigerate any leftovers.