In a stand-up mixer beat together the butter and sugars on medium speed for 5 minutes.
Add the egg and chocolate mint milk and continue mixing until combined.
Slowly add the flour, cocoa, baking soda and salt mixing on low speed until incorporated.
Cover a baking sheet with aluminum foil or parchment paper. Form dough into walnut sized balls. Place 2 inches apart on a baking sheet and press down slightly.
Cover with cling film and chill in the refrigerator for at least 2 hours or up to 48.
Bake cookies for 10-12 minutes until the tops are just set.
Remove from oven and let cool on their tray for 5 minutes before transferring to a wire rack to continue cooling.
To make cocoa icing: In a small saucepan combine sugar, cocoa, butter, and chocolate mint milk. Stirring occasional with a metal whisk bring to a slow boil for 1 minute. Remove from heat and let cool for 5 minutes. Beat with a whisk or electric mixer until icing no long has a glaze look.
Once cookies have cooled, you can ice with cocoa icing and top with crushed candy cane peices.