Whisk together flour, baking powder, and salt. In a large bowl, with an electric mixer on medium, beat butter and sugar until creamy. Add egg and vanilla. Beat in flour mixture on low until combined, scraping down sides of bowl. Divide dough in half; form into disks and wrap in plastic. Chill 2 hours or overnight.
Heat oven to 350°F. Roll out dough on a lightly floured surface to a 1/8-inch thickness. Cut out with cookie cutters. Reroll scraps.
Bake on ungreased baking sheets 7 to 9 minutes, until edges are golden. Cool on sheets 5 minutes; then cool on racks. Decorate as desired.