Preheat the oven to 300° F.
Mix together the butter, sugar, and vanilla with an electric mixer.
Blend in the crushed peppermint candy.
Gradually blend in flour and cornstarch.
Roll into 1″ balls, and place on a greased cookie sheet.
Using a glass dipped in powdered sugar, press down on each cookie to flatten it.
Bake in the oven for 25 to 30 minutes, until the bottom begins to brown.
Remove cookies from oven and allow to cool a little for 5 minutes.
Mix all frosting ingredients, except peppermint candy in a mixing bowl.
Drizzle the frosting over the center of each cookie.
Sprinkle the crushed peppermint candy on top of the frosting.
Allow to completely cool.