1. To prepare cookies, spoon flour into dry measuring cups; level with a knife. Combine flour, baking powder, baking soda, and salt, stirring with a whisk.
2. Place granulated sugar and butter in a large bowl; beat with a mixer at medium speed until well blended (about 5 minutes). Beat in 1 teaspoon vanilla and egg. Add flour mixture, beating at low speed until blended.
3. Place dough between two sheets of plastic wrap. Roll dough to a 1/4-inch thickness. Chill 1 hour.
4. Preheat oven to 375°.
5. Cut dough with a 2 1/4-inch egg-shaped cutter. Place cookies on a baking sheet lined with parchment paper. Bake at 375° for 8 minutes or until edges of cookies are browned. Cool cookies 1 minute on pan. Remove cookies from parchment; cool completely on a wire rack.
6. To prepare icing, combine powdered sugar, milk, and 1/4 teaspoon vanilla; stir until smooth. Add food coloring, if desired. Stir well. Spread or pipe icing onto cookies.