Whisk the flour, baking powder, and salt in a medium bowl.
In a stand mixer fitted with the paddle attachment (or in a large mixing bowl with a hand-held electric mixer), beat the butter and sugar on medium speed until light and fluffy, 3 to 4 minutes. Scrape down the bowl and beater with a rubber spatula as needed.
Add the egg and vanilla and continue to mix until well blended, about 1 minute. Reduce the speed to low, add the flour mixture, and mix until well blended, about 1 minute more.
Divide the dough into thirds and form each piece into a 5-inch square. Wrap in plastic and chill until firm, about 30 minutes. Using a rolling pin, roll a dough square between 2 pieces of parchment to about 1/8 inch thick and 11 inches square. Slide the dough and parchment onto a cookie sheet and refrigerate; repeat with the remaining dough, stacking the squares sandwiched in parchment on top of each other on a single cookie sheet, and refrigerate until very firm, about 45 minutes.
Position racks in the upper and lower thirds of the oven and heat the oven to 350°F. Line 2 or more cookie sheets with parchment or nonstick baking liners.
Working with one square of dough at a time, remove the top piece of parchment and cut out shapes with a floured cookie cutter. Arrange them about 1 inch apart on the prepared sheets. Freeze until very firm, about 20 minutes.
Bake two sheets at a time, rotating and switching their positions halfway through, until the edges and bottoms of the cookies are golden (or darker brown for chocolate cookies), about 11 minutes. Meanwhile, press the dough scraps together, re-roll, chill, and cut more shapes; freeze. Repeat until all of the dough is used. Let the cookies cool on their sheets on wire racks for about 5 minutes and then transfer to the racks to cool completely.
Serve, decorate, or freeze the cookies.
You can refrigerate the rolled-out dough, wrapped well in plastic, for up to 3 days or freeze for up to 1 month. Thaw at room temperature for 10 to 15 minutes before cutting out shapes.
Baked, undecorated cookies can be frozen in an airtight container for up to 1 month; thaw in the refrigerator before decorating.
Decorated cookies can be kept at room temperature for up to 3 days.