Heat butter, stirring constantly, in a saucepan over medium-low heat until it browns and becomes very aromatic, 5 to 10 minutes. Whisk confectioners’ sugar into butter until dissolved and remove saucepan from heat; stir in milk and vanilla extract.
Beat butter mixture using an electric mixer until frosting is smooth; add more milk if frosting is too thick.
Make sure to stir butter continuously while heating and be very careful to not burn the butter – if you do, start over again with new butter.
If you are not using the frosting immediately, cover with plastic wrap and refrigerate so it does not dry out.
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