Preheat the oven to 350°F and grease a 9x9 inch metal baking pan.
In a medium bowl, whisk together flour, baking powder, baking soda, and salt. Set aside.
Whip the 10 tablespoons of brown butter until pale and fluffy.
Add sugars, eggs, and vanilla, mixing until smooth.
Alternate adding buttermilk and dry ingredients to the batter, mixing until smooth.
Pour batter into the prepared pan and bake for 35-38 minutes. Let cool completely.