Melt butter in a saucepan over medium-low heat. Increase the heat slightly once melted, stirring frequently. The butter will bubble and pop as water evaporates. Once this slows, watch closely as the milk solids begin to brown. Stir a few more times and remove from heat once you see browned bits at the bottom.
Graham Cracker Cookies:
Transfer the browned butter into a large bowl and let it cool for 10 minutes. Preheat the oven to 350°F and line a baking sheet with parchment paper. Crush the graham crackers into fine crumbs using a zip-top bag and a rolling pin.
Add both sugars to the cooled brown butter and stir until fully combined, about 1-2 minutes. Mix in the egg and vanilla extract until well blended.
Incorporate the flour, graham cracker crumbs, baking soda, and salt into the butter mixture, mixing thoroughly. Fold in the milk chocolate chips.
Use a ⅓ cup measure to scoop out 8 cookie dough portions, placing them on the prepared baking sheet. Bake for 10 minutes until the cookies are set but not fully golden at the edges. Remove from the oven and press two marshmallow halves onto each cookie. Set the oven to broil on high.
Broil the cookies for 1-2 minutes, watching closely until the marshmallows are lightly browned. Cool the cookies for 10-15 minutes before adding the chocolate drizzle and graham cracker crumbles.
Chocolate Drizzle & Graham Cracker Crumbles:
Melt 1 tablespoon of butter in the microwave, then stir in the chocolate chips until fully melted. Set aside.
Melt 2 teaspoons of butter and mix in the finely ground graham cracker.
Drizzle the melted chocolate over the warm cookies using a spoon and sprinkle with the buttered graham cracker crumbs. Enjoy!