Whisk together the flour, baking powder and salt in a small bowl.
Beat the superfine sugar and butter with an electric mixer on medium speed in a large bowl until light and fluffy, about 5 minutes; beat in the egg, then the vanilla. Add the flour mixture and mix on medium-low speed until completely incorporated. Divide the dough in half, pat into 2 discs about 1/4 inch thick, wrap in plastic wrap and refrigerate until firm, about 1 hour.
Position oven racks in the top and bottom thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper.
Let the dough sit at room temperature for a few minutes to make rolling easier. Roll out 1 disc of dough at a time between 2 pieces of parchment paper until 1/8 inch thick. Cut out shapes with cookie cutters and arrange about 2 inches apart on the prepared baking sheets.
Bake until the cookies are golden brown on the bottom, 10 to 12 minutes. Let the cookies cool on the baking sheets until firm enough to transfer to a wire rack; let cool completely.
Gently gather any scraps of dough into a ball and press into a disc; wrap in plastic wrap and refrigerate the disc until firm enough to roll, about 1 hour. Cut out as many cookies as possible and bake.
