In a microwave-safe bowl, microwave the butter and chocolate at 30-second intervals, stirring well until smooth. Set aside.
⅓ cup unsalted butter, 1 cup chopped semisweet chocolate
In a medium bowl, whisk together the flour, cocoa powder, baking powder, and salt.
1¼ cup all-purpose flour, 3 tablespoons unsweetened cocoa powder, 1 teaspoon baking powder, ¼ teaspoon kosher salt
In a large bowl, use a hand mixer to beat the sugar, eggs, and vanilla extract until frothy.
1 cup granulated sugar, 2 large eggs, 1 teaspoon pure vanilla extract
Add the chocolate mixture and mix until smooth.
Gradually add the dry ingredients and mix until just combined. Wrap the dough in plastic wrap and place it in the refrigerator for at least 1 hour.
Preheat oven to 350°F. Line a baking sheet with parchment paper.
Roll the dough into balls. Place on the baking sheet at least 2 inches apart. Flatten the balls slightly. Bake the cookies for about 10 minutes.
While cookies are baking, cut the marshmallows in half. Once cookies are baked, place a marshmallow (cut side down) into the center of each cookie and lightly press down. Place cookies back into the oven for another 2-3 minutes.
Transfer cookies to a wire rack to cool completely.
For the chocolate icing, combine all icing ingredients except for the sprinkles in a bowl and mix together with a whisk.
½ cup powdered sugar, 1 tablespoon unsalted butter, 1 teaspoon unsweetened cocoa powder, 2 tablespoons hot water
Place the wire rack with cookies over a baking sheet. Spoon a small amount of icing onto the top of each marshmallow. Top with sprinkles.