Whisk flour, baking powder, and salt in a medium bowl. Cream butter and sugar with a mixer until light and fluffy. Beat in egg and vanilla, until just combined. Reduce speed to low, and slowly add flour mixture, beating until just combined. Shape into a disk, wrap in plastic, and refrigerate until firm, about 30 minutes (or up to 2 days).
Preheat oven to 325 degrees. Cut dough in half. On lightly floured parchment, roll out each piece slightly thicker than 1/8 inch. Using scallop-edged or fluted cutters, cut out cookies, and place on a baking sheet 1 inch apart. Freeze until firm, about 10 minutes.
Remove from freezer, and cut out doily patterns in centers of cookies using aspic cutters; discard scraps. (If dough softens, return to freezer until firm.) Bake until edges begin to turn golden brown, 12 to 15 minutes. Let cool on a wire rack.
Cookies will keep, covered, for up to 5 days.