For the Cake: Brown the Butter: Melt 1 cup butter in a medium saucepan over medium heat, swirling until it turns golden brown and has a nutty aroma. Let it cool slightly.

1. Prepare the Dry Ingredients: Whisk together flour, baking powder, and salt. Set aside.
2. Mix the Wet Ingredients: Beat the brown butter and sugar until smooth. Add eggs one at a time, then stir in vanilla extract.
3. Combine Wet and Dry Ingredients: Gradually add dry ingredients to the wet mixture, alternating with milk. Mix until just combined.
4. Bake the Cake: Preheat oven to 350°F (175°C). Grease two 9-inch round cake pans. Divide batter evenly between pans and bake for 25-30 minutes or until a toothpick comes out clean. Let cool for 10 minutes in the pans before transferring to a wire rack.
5. For the Frosting: Brown the Butter for Frosting: Melt 1 cup butter in a saucepan, then continue cooking until it turns golden brown. Let cool.
6. Make the Frosting: Beat together the browned butter and softened butter. Gradually add powdered sugar, vanilla, and a pinch of salt. Add cream or milk until desired consistency is reached.
7. Assemble the Cake: Once cakes are completely cool, frost the first layer, then top with the second layer. Frost the entire cake and garnish with additional brown butter drizzle or toasted nuts if desired.