In a stand mixer, cream together butter and confectioners' sugar. Add sour cream, peppermint extract flour, and cornstarch.
Chill dough in the refrigerator for 30 minutes.
Remove the dough from the fridge and preheat the oven to 350℉.
Line a baking sheet with parchment paper. Use a small cookie scoop to scoop dough and form into equal-sized balls. Place the cookie dough balls on the pan spaced 1" apart.
Bake in the oven for 8-10 minutes until the tops are set and spring back when lightly touched. Place on a cooling rack.
Icing
In a bowl, whisk together butter, milk, confectioners' sugar, peppermint extract, and 1-3 drops of pink icing gel or food coloring.
Dip each cooled cookie in the icing and sprinkle with crushed candy canes.