1. Cook butter in a small heavy saucepan over medium heat, stirring constantly, 6 to 8 minutes or until butter begins to turn golden brown. Remove pan from heat immediately, and pour butter into a small bowl. Cover and chill 1 hour or until butter is cool and begins to solidify.
2. Beat butter at medium speed with an electric mixer until fluffy; gradually add powdered sugar alternately with milk, beginning and ending with powdered sugar. Beat mixture at low speed until well blended after each addition. Stir in vanilla.