Cream butter and confectioners' sugar until light and fluffy. Beat in vanilla and peppermint extracts. In another bowl, whisk together flour, cornstarch, and salt. Gradually add the flour mixture into the butter mixture until well combined. Cover and refrigerate for 30 minutes.
Preheat oven to 350°F. Scoop cookie dough by the teaspoon and place 2 inches apart onto a parchment lined baking sheet. Bake until bottom of cookies are light brown, about 9 - 12 minutes. Remove from baking sheet to wire racks and allow to cool completely.
For the frosting, combine all ingredients in a medium bowl and beat at medium speed, scraping the bowl often, until fluffy, about 3 minutes. Spread the frosting over the cooled cookies and sprinkle with crushed candies. Store in an airtight container.