Whisk together flour, baking powder, and salt in a large bowl.

Cream butter and sugar with a mixer on medium speed until pale and fluffy. With mixer running, add egg and vanilla. Reduce speed to low. Add flour mixture gradually, beating until just incorporated.

Divide dough in half; flatten each half into a disk, and wrap each in plastic. Refrigerate until firm, at least 1 hour (but preferably overnight).

Preheat oven to 325 degrees with racks in top third and lower third. Let 1 disk of dough stand at room temperature just until soft enough to roll, about 10 minutes. On a lightly floured work surface, roll out dough to just under 1/4-inch thickness, adding more flour as needed to keep dough from sticking. Cut out cookies with a 2-inch heart cutter, and place them 2 inches apart on a parchment-lined baking sheet. Roll out scraps once, and repeat. Repeat with remaining disk of dough. Freeze cookies until very firm, about 15 minutes.

Bake cookies, rotating sheets halfway through and switching from top to bottom, until edges turn golden brown, 14 to 16 minutes. Let cool on sheets on wire racks.

Once frosted, let the cookies stand uncovered overnight until dry.

Cookies will keep, covered, for up to 1 week.