1. Preheat oven to 350°F.
2. Put flour in a bowl and make a well in center.
3. Beat together butter and sugar in another bowl with an electric mixer until light and fluffy, then beat in egg and vanilla until blended.
4. Add butter mixture to flour and rub it into flour with your fingertips just until dough comes together in a ball (do not overwork dough, or it will be tough).
5. Halve dough and work with 1 half at a time. Roll out dough 1/4 inch thick on a floured surface with a lightly floured rolling pin.
6. Cut out cookies with a floured cutter (we used a 2 1/2-inch fluted round cutter), arranging cookies, as cut, 1 inch apart on ungreased baking sheets. Reroll scraps once, using as little flour as possible, and cut out more cookies.
7. Bake cookies, 1 sheet at a time, in middle of oven until very pale golden, 12 to 15 minutes.
8. Cool on baking sheet 1 minute, then transfer to a rack to cool 5 minutes if serving warm, or to cool completely.

Makes about 3 1/2 dozen (3-inch) cookies.

Make ahead: cookies keep in an airtight container 1 week.