Start by browning your butter. This is optional, but it does taste phenomenal. In a sauté pan, heat your butter over low until melted. Continue cooking until the milk solids sink to the bottom and turn amber-brown. Move to a heat-proof bowl, and chill for 30 minutes. It will still be liquid but cool to the touch.
In the bowl of your stand mixer, combine the brown sugar and brown butter. Whip until creamy, about 5 minutes, stopping occasionally to scrape down the sides of the bowl and the paddle.
Add in the vanilla, milk, and egg. Beat until smooth. Scrape down the sides every now and then to make sure it's fully smooth.
Add in the flour, salt, baking soda, chocolate chunks, and graham cracker bits. Stir until fully combined. The mixture will be slightly crumbly but will hold its shape when squished together. Move to a container with a lid and refrigerate for at least 4 hours and up to 48 hours.
Divide the dough into 8 portions. Flatten each one slightly in your hand. Place a marshmallow in the center, and form the dough back around it. Pinch and smooth any seams to discourage excess marshmallow leakage. Some is fine on top, but you want to avoid leaks on the bottom.
Repeat until you have used all the dough and marshmallows.
Place the cookie dough balls back in the fridge while you preheat the oven to 350ºF.
Set the cookies on a lined baking sheet. There should be 2 inches between each cookie.
Bake for 15-20 minutes. Allow to cool for 5 minutes to set on the baking sheet before transferring to a wire cooling rack. They are best served warm!