In a mixing bowl, whisk together the flour, cornstarch, and salt. Set aside.
In the bowl of an electric stand mixer fitted with the paddle attachment, cream together the butter and powdered sugar until smooth and well combined.
Mix in the peppermint extract. Then, on low speed, slowly add in the flour mixture, mixing just until incorporated.
Cover the bowl with plastic wrap and chill the dough in the fridge for about 1 hour, until nearly firm. Preheat the oven to 350°F (175°C) during the last 15 minutes of chilling.
Scoop the dough, one tablespoon at a time, and shape it into a ball. Place on an ungreased 18×13-inch baking sheet, spacing the cookies 2 inches apart. Keep the dough you aren’t working with chilled.
Bake in the preheated oven for 10-13 minutes, or until the cookies are set and the bottoms are lightly golden brown. Allow the cookies to cool on the baking sheet for at least 10 minutes, then transfer them to a wire rack to cool completely.
For the frosting, in the bowl of an electric stand mixer, cream together the butter, half and half, peppermint extract, and powdered sugar until light and fluffy.
Working one cookie at a time, spread frosting over the cooled cookies, and immediately sprinkle crushed peppermint bits on top before the frosting sets. Repeat for all cookies.
Store the cookies in an airtight container at room temperature.