In a stand-up mixer beat together the butter and sugars on medium speed for 5 minutes.
Add the egg, chocolate milk, peppermint extract, mixing until combined.
Slowly add the flour, cocoa, baking soda and salt, blending on low until incorporated.
Cover a baking sheet with parchment paper. Roll dough into small balls, then place 2 inches apart on baking sheet, pressing each down slightly.
Cover pan with plastic wrap and chill in refrigerator for at least 2 hours.
Bake cookies for 10-12 minutes.
Remove from oven and let cool for 5 minutes before transferring to a wire rack. Let cool completely before icing.
Icing
In a small saucepan on medium heat, combine sugar, cocoa, butter and chocolate milk. Stir with a whisk, bringing to a boil for 1 minute. Remove from heat and let cool for 5 minutes. Add peppermint extract then beat in an electric mixer until icing no long has a glaze look.
Using a decorating spatula, top cookies with icing and then garnish each with crushed candy cane pieces. Chill to set fully. Store in an airtight container.