Heat oven to 350°F. In large bowl, break up cookie dough. Stir or knead in cherries. Cover; freeze dough 30 minutes for easier handling.
Shape dough into balls, using 1 teaspoon dough for each, and roll in sugar; place 2 inches apart on ungreased cookie sheets.
Bake 9 to 12 minutes or until set and edges are light golden brown. Cool 1 minute; remove from cookie sheets and place on wire racks. Cool completely, about 10 minutes.
In small microwavable bowl, microwave chocolate chips and shortening on High 1 minute, stirring once halfway through microwaving, until melted. If necessary, continue to microwave on High in 15-second increments, stirring until smooth. Drizzle icing over cooled cookies.