In a small saucepan, melt the butter on the stovetop.
When melted, transfer to a large mixing bowl, then add cocoa powder and sugar. Mix well.
In a smaller mixing bowl, combine flour, salt, baking soda, and espresso powder. Whisk to combine.
Add egg and vanilla to butter mixture, making sure the butter is cool enough to not cook the egg. Mix well.
Add flour mixture to butter mixture and combine them well.
Place in fridge and let chill 30 minutes.
When ready to bake, preheat the oven to 350 degrees.
Roll dough into balls(using about 1.5tbsp of dough per ball), and roll each ball into the extra sugar.
Line a baking sheet with parchment paper, and place the cookies on the sheet, roughly 2" apart.
Bake for 8-10 minutes, making sure to not over bake. It is difficult to tell, because the dough is already brown, but your cookies should puff up in the center and have cracks forming.
Allow to cool a few minutes before transferring to a wire rack to cool completely.
Once cookies are cool, add powdered sugar, peppermint extract, and milk to a small bowl, an stir until completely mixed. (You may need to add more or less milk, depending on the consistency.)
Using the back of a spoon, glaze each cookie.
Allow to sit at room temperature for an hour or two, to let the glaze set properly. The glaze should harden, but the timing can differ, based on how thick or thin the glaze is.
Enjoy!