Preheat the oven to 350 degrees. Grease and flour three 8-inch round cake pans.
In a bowl or liquid measuring cup, whisk together the milk, egg whites, and vanilla. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, place the flour, sugar, baking powder, and salt. Stir on low speed to just until combined. Increase the speed to medium and add in the softened butter - a few pieces at a time. Continue to mix until the butter pieces are the size of small peas. Pour in half of the milk mixture and mix on medium-high until light and fluffy, about 1 minute.
Stop the mixer and scrape down the bowl. With the mixer on low, stream in the remaining milk mixture. Mix on medium-low no longer than about 30 seconds or until incorporated. If the batter looks a little lumpy, resist the urge to over-mix. Instead, stop the mixer and give it a few stirs by hand.
Evenly distribute the batter between the prepared pans. Bakes for 22 to 25 minutes, or until a toothpick inserted into the center of the cakes comes out clean. Place the cake pans on a wire rack and let cool for about 10 minutes. Remove the cakes from their pans and continue to cool completely on a wire rack before filling and frosting.