Place butter and chopped chocolate in a heat-safe bowl. Microwave in 30-second intervals, stirring until smooth. Let cool for 15 minutes.
In a medium bowl, whisk together flour, cocoa powder, baking powder, and salt.
In a large bowl, beat brown sugar, eggs, and vanilla on low speed until smooth. Mix in the cooled chocolate mixture until just combined.
Add dry ingredients in two batches, mixing until incorporated. Dough will be thick and sticky. Cover and refrigerate for at least 1 hour.
Preheat oven to 325°F (165°C). Line baking sheets with parchment paper. Scoop tablespoon-sized portions of dough 2 inches apart and bake for 10–12 minutes until crackled.
Remove cookies from oven and press a marshmallow half onto each. Return to oven for 3 minutes until marshmallows soften.
Whisk powdered sugar, melted butter, cocoa powder, hot water, and vanilla until smooth.
Spoon icing over marshmallows, add sprinkles if desired, and allow icing to set completely before serving.
