Preheat the oven to 350° and line an 8-inch-square baking pan with foil so it folds over the sides.
In a food processor, finely grind the graham crackers (you should have 1 1/2 cups). Transfer the crumbs to a medium bowl. In a small saucepan, melt 5 tablespoons of the butter and stir the butter into the crumbs. Press the crumbs evenly into the bottom of the prepared pan and bake for about 12 minutes, until set; let cool completely on a rack.
Meanwhile, in a medium microwave-safe bowl, combine 1 cup of the chocolate chips with the cocoa powder, 3 tablespoons of water and the remaining 5 tablespoons of butter. Microwave at high power for 1 minute, until almost melted, then whisk until smooth. Let cool to room temperature.
In a small bowl, whisk the sugar with the flour, baking powder and salt. Whisk the eggs and vanilla into the cooled chocolate mixture. Stir in the flour mixture and the remaining 1/2 cup chocolate chips; stir to combine.
Scrape the batter over the crust and spread evenly with an offset spatula. Bake for 35 to 40 minutes, until the brownies are set and a toothpick inserted in the center comes out almost clean.
Preheat the broiler to low and set a rack 8 inches from the heat. Top the brownies with the marshmallows and broil until lightly browned, about 2 minutes. Let the brownies cool completely on a rack, then using the foil, lift the brownies out of the pan. Cut into squares and serve.
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