Place a rack in the center of the oven and preheat oven to 350°F. Butter a 9×9-inch baking pan with 2.5-inch-high sides. Line with parchment paper so it hangs over the edges on two sides and lightly grease the paper. The parchment will make the brownies easy to remove from the pan. Set aside. (This recipe can also be made in a 9×13-inch pan for a thinner brownie, cooking for 30 to 35 minutes.)
In a medium bowl, whisk together flour, baking powder, and salt. Set aside. Create a double boiler and melt butter and unsweetened chocolate over simmering water. Stir chocolate and butter in this double boiler until melted and smooth.
In a medium bowl beat together eggs, sugars and vanilla. Stir in warm chocolate mixture, then the dry ingredients. Fold in graham crackers and chocolate chips. Pour batter into prepared pan. Dot with 12 large marshmallows. Bake until toothpick inserted into center comes out with moist crumbs attached, about 45-50 minutes.
Marshmallows will be browned and puffy but will deflate as the brownies cool. Cool for at least 20 minutes than slice with a sharp knife, cleaning the knife with hot water if it gets too messy and sticky. Serve or wrap individually in wax paper for storing.