Preheat the oven to 350°F (175°C). Line four 7-inch or three 8-inch round pans with parchment rounds and grease them with non-stick cooking spray or homemade cake release. I used 7-inch pans for this recipe.
In the bowl of a stand mixer with a paddle attachment (or using a hand mixer), combine 3 cups of flour, 3 cups of sugar, 2 1/2 tsp of baking powder, and 1 tsp of salt. Mix on a low speed until fully combined.
Gradually add half (about 3/4 cup) of the room-temperature brown butter to the dry ingredients on low speed. Continue mixing until no large chunks of butter remain and the mixture resembles moist sand.
Add 1 cup of egg whites on low speed until just incorporated, scraping the sides and bottom of the bowl as needed. Then, mix in 1 1/2 cups of sour cream, 1/4 cup of vegetable oil, and 2 tsp of vanilla extract. Mix on low speed until fully incorporated. Scrape the bowl again with a rubber spatula and beat on medium speed for about 30 seconds to ensure everything is well mixed and to help lighten the texture of the cake layers.
Divide the batter evenly between the cake pans (450g per pan for four layers, or 600g per pan for three layers). Bake for 33-36 minutes or until a toothpick inserted into the center comes out with a few moist crumbs. Rotate the pans halfway through to ensure even baking.
Allow the pans to cool for 10 minutes, then run a small offset spatula around the perimeter to release the cake from the pans. To speed up the cooling process, place the cake layers in the freezer for 30 minutes. Once fully cooled, carefully flip the pans and remove the cake layers.
If desired, use a serrated knife to level the tops and remove any caramelization around the edges. This step is optional but helps with frosting. You can also trim and wrap the layers to freeze if preparing in advance. If you freeze the layers, allow them to thaw for about 15 minutes before assembling the cake. The layers should still be slightly cold, which makes assembly easier.