In a stand mixer, cream together butter and sugar until light and fluffy. Stir in peppermint extract.
Mix together flour and cornstarch. Slowly beat into the butter and sugar mixture.
Refrigerate 30 minutes or until the dough is firm enough to shape into balls.
Preheat oven to 350 degrees and line baking sheets with parchment. Shape dough into 1 inch balls (about 2 t) and place onto prepared baking sheets.
Bake for 10 minutes or until the bottoms are beginning to show a light golden brown. Remove from pans and cool completely on cooling racks.
Prepare the frosting. In a bowl beat the butter until fluffy. Beat in the milk and peppermint. Beat in powdered sugar until smooth.
Spread frosting over the cookies and sprinkle with peppermint candy, lightly pressing into the frosting.
Once the frosting has set, store in a covered container.