Preheat oven to 325°F and line a baking sheet with parchment paper and set aside.
In a small saucepan, brown the butter by cooking on medium heat, stirring occasionally, until the milk solids separate and turn a golden brown colour and the butter smells nutty, about 13-15 minutes. Pour 125g of this browned butter into a large mixing bowl and let cool slightly, about 5 minutes.
Meanwhile, stir together flour, baking powder, baking soda, and salt in a small bowl and set aside.
Using an electric mixer, beat the brown sugar into the browned butter until combined. Then, beat in the egg and vanilla extract until it becomes a light brown colour, about 3 minutes.
Add in the flour mixture and roughly fold together using a spatula until semi-incorporated but still with patches of flour. Fold in the crushed graham crackers and chopped chocolate until incorporated and no dry patches of flour remain. Don't overmix.
Use a 2" cookie scooper to scoop out balls of cookie dough and place them onto your baking sheet, about 2" apart to account for spreading. Press a marshmallow on top of each cookie dough mound, creating an indentation where the marshmallow can sit.
Bake cookies for 9-10 minutes or until the marshmallow is lightly browned on top but the cookies are still very soft. After taking them out of the oven, use a spatula to press the puffed up marshmallows down so that they are almost flush with the cookie surfaces. Let cookies cool on the baking sheet for at least 3-5 minutes before carefully transferring to a cooling rack.