Remove dough from wrapper. Divide dough in half; place in separate medium bowls. Let dough stand at room temperature about 15 minutes.
Add black food coloring to dough in one bowl and red food coloring to dough in remaining bowl. Beat doughs separately until evenly colored. Wrap doughs separately in plastic wrap; refrigerate 1 hour.
Preheat oven to 350°F. Roll black dough on lightly floured waxed paper to 1/4-inch thickness. Sprinkle dough with flour to minimize sticking, if necessary. Cut dough using 3-inch round cookie cutter; place cutouts 2 inches apart on ungreased cookie sheets. Repeat with red dough, cutting out same number of red circles as black circles.
Cut and remove circles from centers of 3-inch cutouts using 2-inch round cookie cutter. Place red 2-inch circles into centers of black 3-inch rings; place black 2-inch circles into centers of red 3-inch rings.
Cut and remove circles from centers of 2-inch cutouts using 1-inch round cookie cutter. Place red 1-inch circles into centers of black 2-inch rings; place black 1-inch circles into centers of red 2-inch rings.
Bake 8 to 11 minutes or until firm but not browned. Cool on cookie sheets 10 minutes. Remove to wire rack; cool completely.