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1
PREHEAT oven to 375 degrees F. Line baking sheets with parchment paper.
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2
SIFT together, in a large bowl flour, baking powder and salt. In another large bowl with an electric mixer beat butter until light, about 2 minutes. Gradually beat in sugar until combined well, about 3 minutes. Beat in egg, milk, and vanilla until combined well.
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3
Gradually add flour mixture, beating until batter is combined well. Divide dough in half. Place each half between two sheets of wax or parchment paper. Roll out each half to 1/4 inch thickness. Chill dough until cold and slightly firm, at least 1 hour and up to 3 days.
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4
Working with 1 portion at a time, peel away and replace 1 sheet of paper. Flip dough over and discard the second sheet. On a lightly floured surface, roll out one disk about 1/4-inch thick. Using cookie cutters cut out shapes and transfer to baking sheets about 1 1/2 inches apart. Roll scraps and cut out more cookies.
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5
REPEAT process using remaining dough.
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6
BAKE cookies, 1 baking sheet at a time, 6 to 9 minutes. Cool cookies on pan on a wire rack 5 minutes before transferring to wire rack to cool completely.
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7
MAKE FROSTING: In a small saucepan whisk together milk and flour until completely smooth. Cook mixture over low heat, stirring constantly, until it comes to a boil and thickens. Remove saucepan from heat and whisk mixture until very smooth. Cool milk mixture until warm, 5 to 10 minutes. In a large bowl with an electric mixer beat butter until light, about 2 minutes. Gradually beat in sugar, about 3 minutes. Gradually beat in milk mixture until combined. Beat in vanilla and beat frosting 30 seconds. Spread frosting over cookies.
