Prep: Preheat oven to 325°F. Line a baking sheet with parchment paper and set aside.
Brown butter: 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 for about 10-15 minutes.
Dry mixture: Meanwhile, stir together flour, baking powder, baking soda, and salt in a small bowl and set aside.
Wet mixture: Add the brown sugar to the browned butter and whisk together until combined. Add the egg and vanilla extract and whisk until the mixture becomes a light brown colour, about 3 minutes.
Combine wet and dry mixtures: Add in the flour mixture and fold together using a spatula until semi-incorporated but still with patches of flour. Add the crushed graham crackers and chopped chocolate and finish folding until incorporated and no dry patches of flour remain. Don't overmix.
Scoop cookie dough: Use a 4-tablespoon cookie scooper to scoop out balls of cookie dough and place them onto your lined baking sheet, leaving about 2" in between each one to account for spreading. Firmly press a marshmallow on top of each cookie dough mound, creating an indentation where the marshmallow can sit.
Bake: Bake cookies for 9-10 minutes or until the marshmallows are puffy and lightly browned on top but the cookies are still very soft. After taking them out of the oven, use a fork to push in any uneven edges or swirl the cookies around inside a large round cutter for perfectly round cookies (optional). Let the cookies cool on the baking sheet for at least 5 minutes before carefully transferring them to a wire rack to cool completely or enjoy warm.