Sift flour, baking powder, and salt together into a bowl. Beat butter and sugar with a mixer on medium-high speed until pale and fluffy. Beat in egg and vanilla. Reduce speed to low. With machine running, gradually add flour mixture; mix until combined. Shape into a disk, and wrap in plastic. Refrigerate until firm, at least 45 minutes (or up to 3 days).
Preheat oven to 325 degrees. Transfer dough to a lightly floured work surface. Roll to 1/8 inch thick. Using a 5-inch-tall bunny-shaped cookie cutter, cut out cookies. Space 2 inches apart on parchment-lined baking sheets. Reroll scraps, and cut out. Refrigerate until firm, about 15 minutes.
Bake until golden, 12 to 15 minutes. Let cool completely on sheets on wire racks. Cookies can be stored in airtight containers at room temperature up to 1 week.