Preheat the oven to 350 degrees F. Lin the bottoms of three 8-inch round baking pans with parchment paper and spray with nonstick cooking spray.
In a medium bowl, whisk together the egg whites, egg, buttermilk, and vanilla.
In a stand mixer fitted with a paddle attachment, combine the flour, sugar, baking powder, and salt.
Add the butter one piece at a time, while the mixer is running on low speed, waiting 10 seconds between each addition.
Once all the butter is added, pour in the buttermilk and egg mixture in 3 batches, mixing on medium-low speed between each addition for about 1 minute and scraping the sides of the bowl. The batter should be light and fluffy. Once all of the liquid has been added, fold the batter once or twice to make sure the batter at the bottom of the bowl is getting mixed in and mix for another 20 to 30 seconds more.
Divide the batter evenly between the pans. If you have a kitchen scale, it will be about 16 ounces of batter for each 8-inch round baking pan.
Bake for 20 to 25 minutes, until a cake tester or toothpick inserted in the middle of the cakes comes out clean with just a few crumbs attached.
Cool the cakes in the pans for 10 minutes before inverting them onto cooling racks and removing the parchment paper from the bottoms. Flip the cakes back over and cool completely on the wire racks.
Level the tops of the cakes, if needed, before wrapping each cake layer with plastic wrap and chilling in the freezer for at least an hour before frosting. The cake layers can be made and frozen a few days before assembling a cake if they are wrapped once in plastic wrap. If you are making them more than 2-3 days in advance, double wrap them in plastic wrap, then wrap in foil and seal in a freezer-safe ziptight bag.