Adjust oven rack to middle position and preheat oven to 350°F. Spray two 9-inch round cake pans with nonstick cooking spray, then line the bottoms of each pan with parchment paper, spray with nonstick cooking spray (that contains flour) again, along the bottom of the pan and the sides. If you don’t have cooking spray that contains flour, then grease your pan with butter and dust with flour.
In a bowl whisk the flour, baking powder, baking soda, salt, and 1 ½ cups sugar.
In a second bowl whisk the buttermilk, butter, whole eggs, egg yolks, vegetable oil, and vanilla until smooth.
In the bowl of a stand mixer fitted with a whisk attachment mix the egg whites and cream of tartar on medium speed for 1 minute, or until mixture is foamy. Increase the speed to medium-high and whip until mixture is soft billowy mounds. With mixer on low, gradually add the remaining ¼ cup of sugar, once added turn the speed to medium-high and whip stiff, glossy peaks form, 2-3 minutes. Transfer the egg whites to a 3rd bowl.
In the now empty stand mixer bowl add the flour mixture. With the paddle attachment and the mixer on low, slowly add the buttermilk mixture and mix until incorporated, about 30 seconds, stopping halfway through mixing to scrape down the sides of the bowl with a rubber spatula.
Using a rubber spatula, fold in one-third of the whipped egg whites. Once incorporated add the remaining egg whites and fold until no white streaks remain.
Divide the batter evenly amongst the prepared cake pans. Level the tops with an offset spatula, then gently tap the cake pan on the counter to allow the batter to settle. Bake in preheated oven for 20-22 minutes or until a toothpick inserted in the center comes out with a few moist crumbs. Remove from the oven and let cool in the cake pan for 10 minutes before turning the cakes onto a wire rack to cool completely (about 1.5-2 hours).