Preheat oven to 350°F. Mix flour and nutmeg in medium bowl. Set aside. Beat butter, shortening, sugar and extracts in large bowl with electric mixer on medium speed until light and fluffy. Gradually stir in flour mixture until well mixed. Dough should be slightly moist. If dough looks dry, sprinkle with water and work into dough. If dough is sticky, sprinkle with additional flour.
Spoon dough by teaspoons and shape into balls. Place on ungreased baking sheets. Gently flatten each ball with palm of hand. Press a cherry piece in the center of each cookie.
Bake 20 minutes or until golden. Cool on baking sheets 1 minute. Remove to wire racks; cool completely.