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 immediately from heat. Pour butter into a bowl. Cover and chill 1 hour or until butter is cool and begins to solidify.
Beat butter at medium speed with an electric mixer until fluffy; gradually add powdered sugar alternately with milk, beginning and ending with sugar. Beat at low speed until well blended after each addition. Stir in vanilla.