In the bowl of a stand mixer, cream the butter, sugars and vanilla with the paddle on medium speed until light and fluffy. Add the egg and scrape the bowl and beat until fluffy again.
In a large bowl, whisk together the flour, cocoa, salt and baking soda. Add to the mixer and mix on low until combined. Add the 2 tsp of milk or water and mix again until combined. Chill the dough for an hour to make it easier to work with.
Preheat the oven to 350 and line baking sheets with parchment. Using a cookie scoop, scoop out about tbsp size balls of dough and briefly roll them in your hands to get them round. Leave at least 3 inches between the balls.
Bake for 4 minutes, then top each cookie with a marshmallow half and then rotate the baking sheet and bake 2-3 more minutes until just set and the marshmallow should be melted a bit and spread out.
Remove and cool on the sheet for 5 minutes then move to a cooling rack to full cool before topping with chocolate.
Melt the chocolate in 20 second bursts in the microwave, stirring after each, until it is smooth. Stir in the peppermint extract and spoon over the top of the marshmallow. Top with sprinkles if desired and allow chocolate to fully harden.