In a medium saucepan, using a whisk, blend 1 cup milk into 3 tablespoons all-purpose flour. Cook and stir over medium heat until thickened and bubbly. Reduce heat; cook and stir 2 minutes more. Cover surface with plastic wrap. Cool to room temperature (do not stir).
In a mixing bowl beat 1 cup softened butter, 1 cup granulated sugar, and 1 teaspoon vanilla with an electric mixer on medium speed until light and fluffy. Add the cooled cooked mixture, 1/4 cup at a time, beating on low speed after each addition until smooth. Makes 12 servings.