In a small saucepan heat butter over low heat until melted. Continue heating until butter turns a delicate brown. Remove from heat. In a large mixing bowl combine powdered sugar, 4 tablespoons of the milk, and the vanilla. Add browned butter. Beat with an electric mixer on low speed until combined. Beat on medium to high speed, adding additional milk, if necessary, to reach spreading consistency. This frosts the tops and sides of two 8- or 9-inch cake layers.