In a small saucepan, melt butter over medium-high heat until nut-brown in color, about 10 minutes. Remove pan from heat, and pour butter into a bowl, leaving any burned sediment behind.
Add sugar, vanilla, and 1 tablespoon milk; stir until smooth. If the icing is too thick, add the remaining tablespoon milk, a little at a time, until consistency is spreadable. Let cool 5 minutes. Use immediately.