In a mixing bowl, whisk together the flour and cornstarch, then add the salt. Set aside.
In the bowl of an electric mixer fitted with a paddle attachment, beat the butter and powdered sugar until well combined.
Add the peppermint extract and mix on low speed. Gradually add the flour mixture and mix until just combined.
Cover the bowl with plastic wrap and refrigerate for about 1 hour, until the dough is almost firm. Preheat the oven to 350°F during the last 15 minutes of chilling.
Scoop 1 tablespoon of dough at a time, roll it into balls, and place them on an ungreased 18x13-inch baking sheet, spacing the cookies 2 inches apart. Keep the remaining dough chilled.
Bake for about 12 minutes, until the cookies are set and the bottoms are lightly golden. Let them cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.
For the frosting, beat the butter, half-and-half, peppermint extract, and powdered sugar in the bowl of a mixer until light and fluffy.
Frost each cookie, then immediately sprinkle crushed peppermint on top before the frosting sets. Repeat for all the cookies.
Store the cookies in an airtight container at room temperature.