Line a rimmed baking sheet with parchment paper. Set aside.
In a large mixing bowl, using a handheld mixer on medium speed, cream together the butter and powdered sugar for 1 to 2 minutes or until light and fluffy. Add the peppermint extract and kosher salt. Mix for another 30 seconds until incorporated.
In a small bowl add the cornstarch and all-purpose flour and whisk together to evenly incorporate. Slowly add this to the butter mixture, while mixing on low speed, just until combined.
Using a 1½ tablespoon (or 1½-inch) cookie scoop, scoop out dough balls and place them onto a plate. Once all the cookie balls are scooped, place the plate into the refrigerator to chill for a minimum of 2 hours.
Preheat oven to 350°F.
Once the dough balls are chilled, place them 2 inches apart on the parchment lined baking sheet. Bake the cookies for 10 to 12 minutes. Allow the cookies to cool on the tray for 5 minutes then transfer them to a wire rack to cool completely. Repeat until all the cookies are baked and cooled.
While your cookies are cooling you can make your frosting.
In a medium-sized mixing bowl, using a handheld mixer on medium-low speed, beat the butter until fluffy. Add the peppermint extract, half & half and 1 cup of the sifted powdered sugar. Beat on low for another 30 seconds then add the remaining powdered sugar one cup at a time until all the powdered sugar is fully incorporated and the frosting is smooth and creamy.
Using an offset spatula, or a butter knife, frost each cookie making sure to not go all the way to the edge of the cookie.
Garnish each cookie with a little sprinkle of crushed peppermint candies.