Preheat the oven to 350°F (175°C).
Grease and line two 8-inch round cake pans with parchment paper.
Whisk together the cake flour, baking powder, baking soda, and salt.
In a large mixing bowl, beat the browned butter, granulated sugar, dark brown sugar, and oil for 3 to 4 minutes until light and fluffy.
Add the eggs one at a time, beating well after each addition.
Mix in the egg yolk and vanilla extract.
In a separate bowl, whisk together the milk and sour cream.
Add the dry ingredients and milk mixture alternately, beginning and ending with the dry ingredients.
Mix just until combined.
Divide the batter evenly between the prepared pans.
Bake for 30 to 34 minutes or until a toothpick inserted into the center comes out clean.
Cool in the pans for 10 minutes, then transfer to a wire rack to cool completely.