Preheat the oven to 350° F.
Melt butter in a small light-colored metal saucepan over medium heat. Cook until the butter foams and steams. Swirl the pan occasionally as the foam subsides and watch closely as the butter turns golden brown and the milk solids drop to the bottom of the pan. Pour the browned butter and butter solids into a small bowl and let stand until cool but not solidified.
Use a pastry brush to coat a 10-inch cast iron pan with a little of the browned butter. Set aside.
Combine the browned butter, sugars, eggs, and vanilla extract in a large mixing bowl. Mix until well incorporated.
Add the flour, graham cracker crumbs, cornstarch and salt. Mix well until thick dough forms. Stir in the chocolate chips.
Press half of the cookie dough in the bottom of the prepared pan. Top with dollops of marshmallow cream. Place remaining cookie dough on top of the marshmallow cream using a spoon or an ice cream scoop. Allow some of the marshmallow cream to remain exposed. Smooth down the dough with a greased rubber spatula or by pressing it down with damp hands (dip hands in water, then press).
Bake for 30 minutes, or until the top of the cookie is well set and golden brown. (The center of the cookie will be gooey, and the edges crisp). Immediately garnish the top of the cookie with graham crackers and squares of the bar chocolate.
Let cool for 10 minutes before serving. Serve warm with vanilla ice cream.