Brown the butter by melting unsalted butter over medium-low heat, swirling until it turns amber color, then cool slightly.

Preheat oven to 175°C (350°F). Grease and line three 20-cm round cake pans with parchment paper. Chop pecans finely.

In a bowl, whisk together cake flour, granulated sugar, light brown sugar, baking powder, baking soda, and salt.

Incorporate cooled browned butter into dry mixture until it resembles coarse sand.

In another bowl, mix buttermilk, eggs, and vanilla extract until combined.

Gradually add wet ingredients to dry mixture and mix until smooth. Fold in chopped pecans.

Divide batter among pans, smooth tops, and bake for 30 minutes until a toothpick comes out clean. Cool layers in pans.

Beat together softened butter, cream cheese, brown sugar, vanilla extract, and salt until creamy. Gradually mix in powdered sugar and adjust consistency.

Assemble the cake by layering frosting between cooled layers and on top, optionally decorate with chopped pecans.