In large bowl, whisk flour, baking powder, and salt. With an electric mixer, cream butter and sugar until light and fluffy. Beat in egg and vanilla. With mixer on low, gradually add flour mixture; beat until combined. Divide dough in half; flatten into disks. Wrap each in plastic; freeze until firm, at least 20 minutes, or place in a resealable plastic bag, and freeze up to 3 months (thaw in refrigerator overnight).
Preheat oven to 325 degrees. Line baking sheets with parchment. Remove one dough disk; let stand 5 to 10 minutes. Roll out 1/8 inch thick between two sheets of floured parchment, dusting dough with flour as needed. Cut shapes with cookie cutters. Using a spatula, transfer to prepared baking sheets. (If dough gets soft, chill 10 minutes.) Reroll scraps; cut shapes. Repeat with remaining dough.
Bake, rotating halfway through, until edges are golden, 10 to 18 minutes (depending on size). Cool completely on wire racks. To ice cookies, spread with the back of a spoon. Let the icing harden, about 20 minutes. Decorate as desired.
Use flour on utensils to keep dough from sticking: Dip the cookie cutters, and dust the spatula before transferring uncooked dough to a sheet. Rolling dough between two sheets of floured parchment will keep it from sticking to the rolling pin.