Preheat oven to 325 degrees. Melt butter over medium heat then reduce to medium-low. Continue heating (low and slow) and stirring for 8-10 minutes total. The butter will foam, solids will form, and you should smell a warm, nutty aroma. Scrape butter and solids into a bowl and set it in an ice bath or in the fridge for about 10 minutes.
Meanwhile, mix flour, baking powder, and sea salt in a bowl. In another bowl, beat brown butter and sugar together then beat in eggs, yolks, vanilla, and milk. Add in flour, scraping bowl as necessary. Divide batter evenly into three greased-and-floured nine-inch round pans. Bake for about 40 minutes or until a toothpick inserted in the center of each cake comes out clean.
While the cakes bake, melt butter for frosting in a small saucepan over medium-low heat. Stir in brown sugar and cream. Bring to a boil then reduce heat and simmer for about five minutes. Remove from heat and sift in powdered sugar, one cup at a time until your desired stiffness is met. Mix well and allow to cool to room temperature before frosting.
Frost cake between layers and on top for the "naked" look of my birthday cake or go full on.