Preheat the oven to 350°F. with a rack in the center position (upper and lower thirds for layers). Coat two 9-inch round cake pans with pan spray and coat lightly with flour, knocking out the excess.
In a small bowl, whisk together the flour, sugar, cocoa, baking powder, baking soda, and salt. In a large bowl, whisk the buttermilk, yogurt, eggs, egg white, and oil until the mixture is smooth. Stir in the beets.Create a well in the flour mixture and thoroughly whisk in the wet ingredients.
Transfer the batter to the prepared pans and bake for about 40 minutes, until the cakesare slightly domed in the center and a knifeinserted near the center comes out clean. Transfer the pans to a wire rack to cool for 20 minutes, then run a knife around the edge and invert the cake directly onto the rack, about 1 hour longer to cool completely.
Directions for frosting:
Using a standing mixer and the paddle attachment, beat the cream cheese and butter on medium for about 1 minute, until the mixture is smooth. Sift in the sugar and cocoa and continue to mix, starting on low and increasing to medium as the sugar is incorporated, until the frosting is completely smooth. Add the vanilla and mix on low until it is incorporated. (Alternatively, use a handheld electric mixer.)
Use an offset spatula to frost the top of the bottom cake first, starting at the center and pushing the frosting out to the sides. Place the second layer on top, spread the rest of the frosting on top of the cake and spread it down and around the sides of the cake.
Cut the cake into wedges to serve. Refrigerate any leftover cake, covered, for up to 2 days.