Preheat the oven to 350°F. Line two baking sheets with parchment paper and set aside.
Whisk together the flour, cocoa, baking soda, and salt in a bowl and set aside.
Using a hand or stand mixer, mix the butter until smooth and creamy. Add the granulated and brown sugars and mix for about 2 minutes, until lightly creamed and fully incorporated.
Add the egg, vanilla, and peppermint extract and mix until combined. Stop the mixer and scrape down the sides and bottom of the bowl.
Add the flour mixture and mix on low speed until just incorporated. It is okay if a few streaks of dry flour remain visible. The dough will be quite thick. Add the chocolate chips and crushed peppermint and mix until combined.
Scoop out 3-tablespoon size portions of dough (best done with a mechanical cookie scoop) and place on a separate piece of parchment paper. Gently flatten the cookie with the palm of your hand. Place a ½ marshmallow in the center.
Lift up the cookie and wrap the edges of the dough around the marshmallow. Do not conceal the marshmallow completely. Form the dough into a ball (it does not need to perfect at all) and roll in the sugar.
Place the cookie on the baking sheet and repeat. Keep the cookies at least 2 inches apart on the baking sheet. Bake for 11 to 14 minutes, or until the edges of the cookies set and the centers remain soft.
Remove the cookies when they are done baking. Promptly take a round cookie cutter (large than the cookies, if possible) and encircle the cookies. Use the inside edges of the cookie cutter to scoot the edges of the cookies into place. This not only helps to form a round cookie shape, but helps encourage those crisp edges and soft, rippley centers.
Cool the cookies on the baking sheet for 10 minutes before moving them to a wire rack. If they are still too soft to move, move the entire piece of parchment paper (with the cookies on top) to the wire rack and continue to cool.
Store leftover in an air-tight container at room temperature for up to a week.