Melt the butter and chocolate together in a medium saucepan over low heat, stirring frequently. Alternatively, microwave the mixture in a microwave-safe bowl at 50% power, stirring intermittently, until fully melted. Set aside to cool slightly.
In a separate bowl, whisk together the flour, unsweetened cocoa powder, baking powder, and salt.
Using an electric mixer, beat the brown sugar, eggs, and vanilla on low speed until well combined. Add the cooled chocolate-butter mixture and mix just until incorporated.
Gradually add the flour mixture, blending until just combined. Scrape down the sides of the bowl, cover, and refrigerate the dough for at least 1 hour, until firm. If refrigerated overnight, allow the dough to sit at room temperature for 30 minutes before shaping.
Preheat the oven to 325°F. Line two baking sheets with parchment paper or silicone baking mats. Scoop tablespoon-sized portions of dough, roll them into balls, and place them on the prepared baking sheets about 2 inches apart. Slightly flatten each ball of dough.
Bake the cookies for approximately 12 minutes. While they bake, slice the large marshmallows in half crosswise. Remove the cookies from the oven and press a marshmallow half, cut side down, into the center of each cookie. Return to the oven for an additional 2–3 minutes. Allow the cookies to cool on the baking sheets for a few minutes before transferring to a wire rack.
Prepare the Icing
To make the icing, whisk together powdered sugar, melted butter, unsweetened cocoa powder, hot water, and vanilla extract in a medium bowl until smooth.
Place the wire cooling rack with cookies over a baking sheet to catch any excess icing. Spoon a small amount of icing over each marshmallow, using the back of the spoon to spread it slightly. Add sprinkles to each cookie immediately after icing to ensure they adhere before the icing sets.
Let the icing firm up for about 30 minutes before serving.
Notes
Store cookies in an airtight container for up to 2 days. Customize the sprinkles to match different holiday themes or occasions.