In a large bowl, with a mixer on medium speed, beat 2/3 cup butter, granulated sugar, and brown sugar until smooth. Add egg and molasses and beat until well blended.
In a medium bowl, mix flour, cinnamon, ginger, cloves, baking soda and salt. Stir into butter mixture, then beat just until dough comes together. Stir in peppermint candy. Cover bowl with plastic wrap and freeze until dough is firm, about 45 minutes.
Shape dough into 1-inch balls and place about 2 inches apart on buttered 12- by 15-inch baking sheets.
Bake cookies in a 350º regular or convection oven until lightly browned, 12 to 15 minutes; if baking two sheets at once in one oven, switch their positions halfway through baking. Immediately, with a wide spatula, transfer cookies to racks to cool. When cool, drizzle with Peppermint Icing; save any remaining icing for other uses.
Peppermint Icing: In a bowl, mix 1 cup powdered sugar, 2 tablespoons milk, and 1/4 teaspoon vanilla until well blended. If too thick to drizzle, mix in more milk, 1/2 teaspoon at a time, until thin enough. Stir in 1 tablespoon finely crushed hard peppermint candy and 1 to 2 drops red food coloring if desired. Makes 1/2 cup.
Nutritional analysis per cookie.