Sift flour, baking powder, and salt into a bowl. In a separate bowl, beat butter and sugar with a mixer on medium speed until pale and fluffy, about 3 minutes. Beat in eggs and vanilla. Reduce speed to low. Gradually mix in flour mixture. Turn out dough, and divide in half. Flatten each half into a disk, and wrap in plastic. Refrigerate until firm, at least 1 hour or overnight.
Preheat oven to 325 degrees, with racks in upper and lower thirds of oven. Let one disk of dough stand at room temperature just until soft enough to roll, about 10 minutes. Roll out dough on a lightly floured work surface to just under 1/4-inch thick, adding more flour as needed to keep dough from sticking. Chill in refrigerator until firm, about 30 minutes. Cut out cookies using desired cutters or templates. Transfer to parchment-lined baking sheets as you work. Roll out scraps, and repeat once. Repeat with remaining disk of dough. Chill cookies in freezer until very firm, about 15 minutes.
Bake cookies, switching positions of sheets and rotating halfway through, until edges turn golden, 15 to 18 minutes. Let cool on sheets on wire racks. Decorate with Royal Icing.
Cookies can be stored at room temperature for up to 1 week.