In a bowl, with an electric mixer on medium speed, beat butter and sugar until smooth. Beat in egg and vanilla until well blended.
In another bowl, mix flour, cornmeal, and baking powder. Stir or beat into butter mixture until well blended.
Divide dough in half and form each portion into a 2-inch-thick log.
Roll waxed paper tightly around each log to wrap airtight. Freeze until dough is firm enough to slice neatly, at least 2 hours, or up to 3 weeks.
Unwrap dough. Using a thin, sharp knife, cut logs crosswise into 1/8-inch-thick slices.
Lay slices 3 inches apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets; if cookies flattened when cut, reshape with your fingers.
Bake cookies in a 350 oven until edges are slightly browned, 15 to 18 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. With a wide spatula, transfer cookies to racks to cool.
