Preheat the oven to 350 degrees F. Grease an 8-by-8-inch baking dish with butter.
In a large bowl combine the butter and brown sugar. Using a hand mixer, blend until fluffy, about 2 minutes. Add the vanilla extract and eggs and mix on medium speed until combined, about 1 minute.
In a separate bowl combine the flour, graham cracker crumbs, baking powder and salt. With the mixer running, slowly add the dry mixture into the wet mixture, and mix until a dough forms.
Transfer the dough to the prepared baking dish and pat out in an even layer to cover the bottom of the pan. Top the dough with the chocolate chips and marshmallows.
Bake until the marshmallows are browned and the dough is cooked through, 20 to 25 minutes. Cool to room temperature, then slice into 2-by-2-inch squares.