Make the brownies:
Preheat the oven to 350ºF. For thick brownies, spray a 9×9″ pan with non-stick spray or line with parchment paper. For thinner brownies, use a 9×13″ pan.
Prepare the brownie mix according to the package directions. Pour the batter into the prepared pan. Bake for 30 minutes, or until almost done. Do not overcook. Remove from the oven and sprinkle marshmallows over the top to cover the brownies evenly. Bake for an additional 3 minutes, or until the marshmallows puff up. Remove from the oven and allow to cool.
Make the frosting:
Combine the butter, powdered sugar, cocoa powder, milk, and vanilla in a large bowl. Beat with electric beaters until smooth. Adjust the consistency with more milk or powdered sugar if needed. Once the brownies are cooled, place spoonfuls of frosting on top of the marshmallows and spread evenly.