PREHEAT oven to 375 degrees F. Line baking sheets with parchment paper.
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.
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.
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.
REPEAT process using remaining dough.
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.
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 confectioners sugar, about 3 minutes. Gradually beat in milk mixture until combined. Beat in vanilla and beat frosting 30 seconds. Spread frosting over cookies.
MAKES about 2 1/2 dozen cookies.