Preheat the oven to 350 degrees F.
With an electric mixer, beat together the butter and sugars in a large bowl until smooth and fluffy, then beat in the eggs. Add the baking soda, vanilla and salt to the butter mixture, mixing just until combined. Add the flour and smashed graham crackers and slowly mix together until combined.
Drop tablespoon-sized dollops of batter 2 inches apart on ungreased baking sheets. Dollop some marshmallow creme onto each of the cookies, and press a piece of chocolate and graham cracker into the marshmallow.
Bake until just set in the center, 7 to 8 minutes. Allow to cool slightly, then transfer to a wire rack to cool completely.