Preheat your oven to 325 degrees F. Prepare three 8-inch or four 6-inch round cake pans by spraying each pan with non-stick spray, covering the bottom of the pan with parchment paper and spraying the parchment paper with the non-stick spray as well. Set aside.
In a medium-sized mixing bowl, combine the whole egg, egg whites, vanilla extract and root beer flavor. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, combine the cake flour, sugar, baking powder and salt. Mix on low speed for about 30 seconds.
With the mixer on low speed, add the butter and root beer. Continue to mix on low until just combined and then increase to medium speed and mix for 1 minute.
Scrape down the sides of the bowl and add the egg mixture in two additions, beating on medium speed after each addition until just combined and all the ingredients are thoroughly incorporated.
Divide the batter into the three prepared cake pans, using a kitchen scale to measure equal parts in each pan. Bake for 25-30 minutes (my cakes took 25 minutes), or until a toothpick inserted into the center of the cake comes out clean.
Let the cakes cool in the pans for 10 minutes and then invert onto wire racks until completely cool.