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.