Preheat oven to 350° F. Lightly grease three 6-inch cake pans and line with parchment paper. (Alternatively, use two 8-inch cake pans). Melt butter and set aside to cool.
In the bowl of a stand mixer, whisk together sugar, eggs, and egg yolks. Set over a pot of barley simmering water (bain-marie)*. Whisk constantly until egg mixture reaches 110° F on kitchen thermometer. Remove from heat and attach to stand mixer fitted with whisk attachment. Whip on medium speed until mixture has increased more than double in volume, about 3-5 minutes.
Sift cake flour over egg mixture. Gently fold to distribute into batter. Take a couple tablespoons of batter and mix it into cooled butter along with vanilla extract. Fold this tempered butter mixture into the rest of the cake batter. Fold until there are no longer any dry streaks of flour.
Evenly distribute cake batter among prepared cake pans, about ⅔ full. Bake for 28-30 minutes until the top of cake is lightly browned, firm to the touch, and toothpick inserted in center of cake comes out clean. Allow cake to cool in pan for 3 minutes. Run a butterknife along the edges of the cake to help release. Unmold and allow cakes to cool to room temperature on a wire rack.
Once cakes have completely cooled, slice each layer in half to end up with 6 layers for 6-inch cake or 4 layers for 8-inch cake.