Preheat oven to 350℉ and grease and flour a pan (either 9x9 inch square or 11x7 inch rectangle.)
In a medium saucepan over medium heat, melt the butter. Remove the saucepan from heat and stir in the chocolate until smooth. Whisk in the sugar, eggs and vanilla extract, then stir in the flour until just blended. Pour mixture into the prepared pan. Use a butter knife to smooth down.
Bake for 20 minutes, then remove from oven. Sprinkle the graham cracker crumbs evenly over the brownies. Spread the marshmallows over the top of the graham cracker crumbs and return to oven. Bake for an additional 5 minutes. Check the brownies with a toothpick until the center comes clear.