In a microwave-safe bowl, heat the butter and chocolate chips in the microwave for 30 seconds. Then stir and heat in additional 15-second intervals, stirring after each, until melted and smooth. Set aside to cool slightly.
In another bowl, whisk to combine the flour, cocoa powder, baking powder, and salt. Set aside.
In a large bowl with a hand mixer, or in the bowl of a stand mixer fitted with the paddle attachment, beat the brown sugar, eggs, and vanilla on low speed until well combined.
Add the cooled chocolate mixture and continue mixing until just combined.
With the mixer on low, gradually mix in the dry ingredients, scraping down the sides and bottom of the bowl as need with a rubber spatula. Mix until just combined.
Cover with plastic wrap and refrigerate the dough for at least 1 hour.
Preheat the oven to 325°F. Line baking sheets with parchment paper. Scoop a tablespoon of dough (or use a medium cookie scoop), shape into a ball, and place on baking sheets 2 inches apart.
Bake for 11 minutes. Remove from the oven and carefully place a halved marshmallow, sticky side down, in the center of each cookie. Return to the oven for 2-3 minutes, until the marshmallows soften and stick to the cookies.
Allow the cookies to cool on the baking sheets for 10 minutes or until they are firm enough to be transferred to a cooling rack.
For the icing, whisk to combine the confectioners' sugar, melted butter, cocoa powder, hot water, and vanilla until smooth.
Place the cooling rack over a lined baking sheet (to catch excess icing). Spoon icing over each marshmallow, spreading slightly with the back of the spoon or butter knife. Immediately add sprinkles. Note: you will want to work quickly before the icing sets.
Allow the icing to set fully (30-60 minutes) before serving or storing.
Notes
You can cover the marshmallows entirely with the icing for a “marshmallow surprise” or drizzle the icing over the cookies to leave some of the marshmallows exposed.Store in an airtight container for 2-3 days at room temperature.