In a large bowl, with mixer at medium speed, cream butter and sugar until light and fluffy. Beat in egg, milk, and vanilla. Blend in flour, baking powder, and salt until well mixed. Divide dough in half and chill 30 minutes.
Preheat oven to 375°F. Grease 2 large cookie sheets. On a lightly floured surface, working with half of the dough at a time, roll 1/8 inch thick. With floured cookie cutters, cut dough into shapes. Using a metal spatula or pancake turner, place cookies 1/2 inch apart on cookie sheets. Reroll trimmings and continue to cut shapes. Bake 8 minutes, or until very lightly browned. Transfer cookies to racks to cool completely.