Beat the dough butter and confectioners’ sugar in a small bowl with a hand mixer or stand mixer for 3 to 4 minutes. Look for a pale, billowy mixture with no dense butter lumps, which shows that the creaming has incorporated air.
Mix the peppermint extract into the butter mixture, then whisk the flour and cornstarch together in a separate bowl and add that mixture gradually. Stop beating when the last dry streaks disappear so the dough stays tender rather than becoming overworked.
Cover the bowl with storage wrap, refrigerate the dough for about 30 minutes, and preheat the oven to 350°F. The chilled dough should feel firm enough to roll without leaving a sticky coating on your palms.
Roll the dough into 1-inch balls and arrange them on ungreased baking sheets with 2 inches between them. Keep the balls similar in size so their centers and edges bake at the same pace.
Bake for 9 to 11 minutes, checking the undersides near the end of the baking window. The bottoms should look lightly golden while the tops remain pale and matte.
Transfer the cookies from the baking sheets to wire racks and let them cool completely to room temperature. The centers and undersides should have no lingering warmth before frosting, or the buttercream may soften and slide.
Whip the frosting butter in a small bowl with a hand mixer or stand mixer, blend in the milk, peppermint extract, and optional coloring, then gradually beat in the confectioners’ sugar. The finished frosting should look smooth and hold a soft mound rather than run off the beaters.
Spread frosting across each cooled cookie and immediately scatter crushed peppermint candies over the top. The fresh frosting should catch the candy pieces before its surface begins to set.