Preheat your oven to 350F. Lightly grease a 10-inch oven-safe cast iron skillet.
In a large bowl, whisk together the melted butter, brown sugar, and granulated sugar until smooth. Add the egg and vanilla extract and whisk until well combined and slightly thick.
Add the flour, graham cracker crumbs, baking soda, and salt. Fold with a spatula until just combined. Fold in 3/4 cup of the chocolate chips.
Press the dough evenly into the prepared skillet. Scatter the remaining 1/4 cup of chocolate chips over the top.
Bake for 20 to 23 minutes, until the edges are golden and set but the center still looks slightly soft. Remove the skillet from the oven.
Immediately scatter the mini marshmallows and broken graham cracker pieces evenly over the top. Return the skillet to the oven and switch to the broiler on low. Broil for 1 to 2 minutes, watching constantly, until the marshmallows are puffed and golden.Remove from the oven and let it cool for 5 minutes before serving warm with scoops of ice cream.