Preheat oven to 350°F. Grease and flour three 6-inch round cake pans. Line with parchment paper and set aside.
In the bowl of a stand mixer, whisk together 2 cups flour (reserve remaining 1/2 cup of flour), sugar, baking powder, and salt. Using a paddle attachment, run mixer on low speed. With the mixer running, add cut butter in three additions. Continue to mix on low speed until mixture is crumbly.
In another bowl, combine milk, eggs, egg white, and vanilla. Add half of liquid mixture to mixing bowl. Mix on medium speed until incorporated. Add remaining half of liquid mixture and mix until batter is throughly combined. The batter may look curdled. Add remaining 1/2 cup flour and mix until batter comes together. Scrape down bowl as needed to ensure thorough mixing.
Evenly distribute batter among prepared cake pans and level using a mini offset spatula or the back of a spoon. Bake for 30-35 minutes until toothpick inserted in center of cake comes out clean. Allow cake to cool in pan for 10-15 minutes. Run an offset spatula along the sides of the cake pan to loosen cake. Invert cake onto a wire rack and allow to cool to room temperature.