In a large bowl, whisk together 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 fluffy. Mix in peppermint extract then with mixer set on low speed slowly add in flour mixture and mix just until combined.
Cover bowl with plastic wrap and chill until nearly firm, about 1 hour.
Preheat oven to 350 degrees F. Line baking sheets with parchment, if desired.
Scoop dough out using a scoop (scant 1½ tablespoons each), shape into a ball then transfer to the baking sheet.
Bake in preheated oven until cookies are set and bottoms are lightly golden brown, about 11-13 minutes.
In the bowl of an electric stand mixer fitted with the paddle attachment, prepare the frosting by mixing the butter, half and half, peppermint extract, and powdered sugar until creamy. Place crushed peppermint candies in a small dish.
Once complete cooled, spread frosting over cookie tops and gently "dunk" into crushed peppermint.