Preheat the oven to 350°F (175°C) and line baking sheets with parchment paper so the cookies release easily and the marshmallow bottoms don’t stick.
Melt the butter and semi-sweet chocolate together in a heatproof bowl over simmering water or in short microwave bursts, stirring until completely smooth. Set aside to cool until warm, not hot.
In a medium bowl, whisk together the flour, cocoa powder, baking powder, and salt until well combined.
In a large bowl, beat the brown sugar and eggs until thick, creamy, and slightly lightened in color, then stir in the vanilla extract.
Slowly add the cooled melted chocolate mixture to the sugar-and-egg mixture, stirring constantly until smooth and uniform.
Gradually add the dry ingredients to the wet mixture, mixing just until no streaks of flour remain; the dough will resemble thick brownie batter.
Shape and Bake
Drop tablespoon-sized mounds of dough onto the prepared baking sheets, spacing them about 2 inches apart to allow for spreading.
Cut each large marshmallow in half and gently press a half, cut side down, into the center of each dough mound so it sits partly nestled in the batter.
Bake for 10–12 minutes, until the edges are set and the tops look slightly shiny and cracked while the centers remain soft.
Remove from the oven and let the cookies cool on the baking sheets for about 5 minutes before transferring them carefully to wire racks to cool completely.
Make the Icing and Decorate
While the cookies cool, whisk together the powdered sugar, melted butter, cocoa powder, hot water, and vanilla extract in a bowl until smooth and pourable, adding a tiny splash more hot water if needed.
Drizzle or spread the icing over the cooled cookies, letting it run slightly over the marshmallow and edges, then immediately top with sprinkles so they stick before the icing sets.
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