Preheat the oven to 350°F. Grease, flour, and line the bottoms of three 6-inch cake pans with parchment paper. Set aside.
In the bowl of a stand mixer fitted with paddle attachment, add the flour, sugar, baking powder, and salt. Mix on low speed just to combine. Add the butter and mix on medium speed until the mixture becomes crumbly and the butter pieces are coated in flour.
Add the vanilla to the milk. Pour in about half of the milk mixture into the butter mixture. Mix on medium speed until evenly moistened. The batter will still be quite lumpy.
Mix the remaining milk with the egg whites. With the mixer on low speed, slowly stream in about ⅓ of the egg mixture at a time. Mix until the egg mixture is completely incorporated then stop the mixer and scrape down the sides of the bowl. Repeat this two more times using the remaining egg mixture. If there are still some lumps in the batter at the end, that is okay. Do not over mix.
Evenly distribute the batter among the prepared pans. Bake for 24 to 27 minutes or until a toothpick inserted in the center of the cakes comes out clean. Allow the cakes to cool on a wire rack for 10 to 20 minutes before removing them from the pans. Let the cakes completely cook before filling and frosting.