Preheat oven to 350°F/175°C. Line four 7-inch or three 8-inch round pans with parchment rounds, and grease with non-stick cooking spray.
Mix 3 cups flour, 3 cups sugar, 2 1/2 tsp baking powder, and 1 tsp salt together in a stand mixer with a paddle attachment or hand mixer until fully combined.
Gradually mix 3/4 cup of room-temperature brown butter into the dry ingredients on a low speed. Continue to mix until no large chunks of butter remain, and the mixture looks like moist sand.
Mix in 1 cup egg whites on low until just incorporated. Scrape the sides and bottom of the bowl as needed with a rubber spatula.
Then mix in 1 1/2 cups sour cream, 1/4 cup vegetable oil, and 2 tsp vanilla extract on a low speed until fully incorporated. Scrape down the sides of the bowl with a rubber spatula one more time, then beat on a medium speed for about 30 seconds to make sure everything is properly mixed together. This also helps lighten the texture of the cake layers.
Divide the batter evenly between the cake pans (600g per pan if you make 3 layers, 450g per pan if you make 4 layers). Bake for 33-35 minutes or until a toothpick comes out with a few moist crumbs. Rotate the pans halfway through to help them bake evenly.
Let the pans cool for 10 minutes, then run a small offset spatula around the perimeter of the pan to separate the cake from the pan.
Place the cake layers into the freezer for 30 minutes to accelerate the cooling process. Once the layers are fully cooled, carefully flip the pans and remove the layers.
These cake layers bake up pretty flat, but I like to use a serrated knife to level the tops and remove any caramelization around the sides of the layers. This step is optional, but it makes the cake a lot easier to frost. You can also trim the layers, then wrap and freeze them if you're making them in advance.
If you make these cake layers in advance and freeze them, let them thaw for about 15 minutes before making the cake. The cake layers should still be slightly cold to the touch, which will make it easier to assemble the cake.