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.
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.
Place dough between two sheets of plastic wrap.
Roll dough to a 1/4-inch thickness. Chill 1 hour.
Preheat oven to 37
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.
To prepare icing, combine powdered sugar, milk, and 1/4 teaspoon vanilla; stir until smooth.