In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar until light an fluffy. About 3-4 minutes.
Add the milk and extracts with the mixer running on low speed. The milk may look curdled.
In a small bowl, whisk together flour, cornstarch, cocoa, and salt.
Gradually add the dry ingredients to the butter mixture on low speed and mix until combined.
On two separate pieces of plastic wrap, shape half of the dough into rolls about 1 1/2-inches thick.
Refrigerate for at least an hour until dough is very firm.
Preheat oven to 375 degrees and prepare a cookie sheet by lining it with parchment.
Remove one of the rolls from the fridge, and with a sharp knife, make slices about 1/4-inch thick. Place on the prepared cookie sheet about an inch apart.
Bake for 13-15 minutes until edges are firm. Transfer cookies to a wire rack to cool completely before dipping them in the chocolate.
Repeat with the second roll of cookie dough roll.
Chocolate Coating
In a microwave-safe bowl, combine chocolate and butter. Heat on high for 30 seconds, stirring until chocolate is smooth and reheating for another 15-30 seconds depending on the consistency.
Mix in the peppermint extract.
Using two forks, dip a cookie into the chocolate, transferring to the rack with the forks.
Place on a cookie sheet lined with parchment paper to set for at least 30 minutes.