Heat oven to 350°F. Work with half of cookie dough at a time; refrigerate remaining dough until needed.
Cut dough into 12 equal pieces. Roll each piece into ball; flatten each until 1/2 inch thick. Place each flattened ball on one end of 1 wooden stick, covering 1 inch of stick. Place stick side down 3 inches apart on ungreased large cookie sheets, overlapping sticks as needed. Repeat with remaining dough and sticks.
Bake 10 to 13 minutes or until edges are light golden brown. Cool 1 minute; remove from cookie sheets. Cool completely, about 15 minutes. Do not pick up cookie pops using sticks until completely cooled.
Stir peppermint extract into frosting until well blended. Frost cookies. Sprinkle with edible glitter. Let stand until frosting is set, about 1 hour. Store between sheets of waxed paper in tightly covered container.