In a large bowl, cream butter and confectioners' sugar until light and fluffy. Beat in egg and extracts. Combine flour and salt; gradually add to creamed mixture and mix well. Divide dough in half; wrap in plastic wrap. Chill overnight or until firm.
On a lightly floured surface, roll out one portion of dough into a 12-in. x 10-in. rectangle, about 1/8-in. thick. Cut into 2-in. squares. In each square, make 1-in. slits in each corner. Bring every other corner up into center to form a pinwheel; press lightly. Sprinkle cookies with red sugar and press a candied cherry piece into the center of each.
Place 1 in. apart on ungreased baking sheets. Bake at 350° for 8-10 minutes. Cool 1-2 minutes before removing to wire racks. Yield: about 4 dozen.