Melt butter over medium-low heat until golden with brown specks and a nutty aroma, then let it cool completely.
Preheat oven to 350°F (175°C) and prepare three 8-inch round pans with grease and parchment.
Whisk cake flour, granulated sugar, brown sugar, baking powder, baking soda, and salt until combined.
Mix cooled browned butter into dry ingredients until crumbly.
Whisk buttermilk, eggs, and vanilla, then gently combine with batter without overmixing.
Fold in finely chopped pecans evenly.
Divide batter into pans and bake about 30 minutes until a toothpick comes out clean.
Cool in pans for 10 minutes, then transfer to a rack until fully cooled.
Beat butter and cream cheese, then mix in brown sugar, vanilla, salt, powdered sugar, and cream until fluffy.
Assemble cake by layering frosting between layers and covering the entire cake, topping with pecans if desired.