Preheat the oven to 350°F. Grease and flour three 8-inch round cake pans and line with parchment circles.
In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, sugar, baking powder, and salt. Mix on low speed until blended. With the mixer running on low speed, add the butter, a few pieces at a time, mixing until the mixture is crumbly.
In a small bowl, whisk together the buttermilk and vanilla. With the mixer running on low speed, pour two-thirds of the buttermilk mixture into the bowl and mix until the batter comes together. Turn the mixer up to medium speed and beat for about 90 seconds, until the batter has gained volume. Stop the mixer and scrape down the sides and bottom of the bowl with a rubber spatula.
Add the egg whites and egg to the remaining buttermilk mixture and whisk together with a fork. Pour half of the egg mixture into the bowl and mix on low speed until some of the liquid has been incorporated, then beat on medium speed to combine completely, 15 to 20 seconds. Stop the mixer, add the remaining egg mixture and beat for an additional 15 seconds on medium speed.
Divide the batter evenly among the prepared pans, smoothing with the back of a spoon until even and bake for 23 to 25 minutes, until a toothpick inserted in the center of the cakes come out clean. Allow the cakes to cool in pans for about 20 minutes before turning them out onto a wire rack. Remove the parchment paper from the bottom of each cake, turn them right side up and leave them to cool completely.
Cake layers will keep at cool room temperature, wrapped tightly in plastic wrap, for up to 24 hours.