In a large bowl combine 1 cup butter, 1/2 cup powdered sugar, and ¾ teaspoon peppermint extract.
Using an electric mixer beat until all combined and creamed together.
Next, add in the flour and cornstarch until well mixed. Refrigerate for 1 hour.
Preheat the oven to 350 degrees.
Shape about 2 teaspoonfuls of dough into balls. Place onto a parchment paper-lined baking sheet.
Bake for 12-15 minutes or until the edges are lightly golden.
Cool for 5 minutes on a baking sheet and then carefully remove to the cooling rack. Be careful as these cookies are fragile. Cool completely.
To make the glaze combine 1 1/2 cups powdered sugar, and 1/4 teaspoon peppermint extract and start by adding 1 tablespoon of milk to start and add more if the glaze is not a drizzly consistency. You don’t want the glaze too thin though.
Drizzle the glaze over the cooled cookies.
Immediately sprinkle with crushed peppermints. Serve.