Stir milk and flour together in a saucepan over low heat; cook, stirring constantly, until thick, about 5 minutes. Cool mixture in refrigerator until cold to the touch, about 15 minutes.
Beat butter, confectioners’ sugar, and vanilla extract together in a bowl until light and fluffy.
Stir cooled milk until smooth. Slowly beat into the butter mixture until frosting is of a spreadable consistency. For a thicker icing, you may not need the entire milk mixture.
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