Preheat oven to 375 degrees and line a baking sheet with parchment paper or silicone baking mats.
Cut cold butter into tablespoon chunks, add to mixer and mix until butter has somewhat softened.
Add in sugars and cream with butter. (Mixture should be smooth, this may take some patience to achieve but there should be no balls/chunks of butter)
Add in eggs and vanilla, mix until well combined.
Sift together dry ingredients, then add to wet ingredients. Mix on low until just combined.
Fold in marshmallows and chocolate chips.
Melt 1/3 C Butter** and dip graham crackers in it. Space graham crackers about 2.5-3 inches apart on a cookie sheet and top with cookie dough mounds.
Bake for 10-13 minutes. Cookie edges will start to set and the tops start to brown. Remove from oven and top with a chunk of chocolate baking bar or candy bar. Let the cookies finish baking on the hot sheet for another 20ish minutes.