Preheat oven to 350 °F. Butter and lightly flour four 6-inch cake pans. Line bottom with parchment paper and set aside.
Sift together cake flour, baking powder, and salt. Set aside. In another bowl, whisk together milk, egg whites, and vanilla. Set aside.
In the bowl of a stand mixer, beat butter until smooth. Add sugar and mix until combined. Scrape down bowl as needed to ensure even mixing.
Add 1/3 of the dry ingredients. Mix on low speed while adding 1/2 of the liquid ingredients. Add another 1/3 of dry, follow with remaining liquid ingredients, and end with remaining dry ingredients. Mix until batter is evenly combined.
Divide batter among prepared cake pans. Bake 30-35 minutes until toothpick inserted in center of cake comes out clean. Let cake cool in pan for 5 minutes before unmolding. Allow cake to cool to room temperature on wire racks.
Once cake is cool, use a serrated knife to level off cake layers if necessary.