MELT BUTTER: Melt the butter in a very large, microwave-safe bowl. Once melted, set aside for 5-10 minutes to cool to room temperature. It's important the butter isn't hot when you add in the sugars or it will melt the sugar and make the cookies greasy.
WET INGREDIENTS: Once butter is COMPLETELY cooled to room temperature, stir in the light brown sugar and white sugar. Whisk until smooth, about 1-2 minutes (the butter and sugars should fully incorporate). Take your time; it may seem like the two won't come together, but keep mixing until completely integrated. Add in one egg and mix to fully integrate. Add in the other egg yolk (only the yolk, discard or save whites for another recipe) and vanilla extract. Mix until completely combined.
DRY INGREDIENTS: In another bowl, toss together the correctly measured flour (Note 1), baking soda, and salt. Add full graham crackers to a blender or food processor and crush until you have 3/4 cup of very fine crumbs (spoon those crumbs into a measuring cup until the measuring cup is over-filled and then level off the top with the back of a butter knife -- See Note 2). Add graham cracker crumbs in and stir dry ingredients until combined.
COMBINE: Combine the wet and dry ingredients and mix until JUST combined (don't over mix). Add in the 1/2 cup milk chocolate chips and 1/2 cup miniature chocolate chips. Stir until chips are integrated into dough. Cover the bowl tightly. Place in the fridge and chill for 30 minutes up to 2 hours.
ROLL DOUGH BALLS & CHILL: Measure out balls of dough (2 tablespoons EACH (41g) in size). Roll into tall cylindrical balls. Place rolled cookie dough balls on a parchment paper lined plate and place in the fridge for another 30 minutes. Roll out all the dough; you should get around 23-24 cookies from this dough.
PREP FOR BAKING: Preheat the oven to 350 degrees F (176 degrees C). Line a baking sheet with parchment paper or use a nonstick liner. Grab 6 cookie dough balls at a time from the fridge and place, very spaced apart, on the cookie sheet.
BAKE: Right before baking, set out the miniature marshmallows and coarsely chop the Hershey's bars. Bake cookies for 8-9 minutes, remove from the oven and, working quickly, immediately press any edges that have strayed out a bit inwards with the back of a metal spatula. Press 3-4 miniature marshmallows into the tops of the cookies and 3-4 pieces of chopped Hershey's chocolate. Return cookies to the oven and bake for another 2-3 minutes or until very slightly browned at the edges and no longer wet looking on top (erring on the side of under baking to keep them soft and chewy.) Let cookies cool on the pan for 5 minutes before transferring to a cooling rack. Repeat with the remaining dough until all the cookies are baked -- or freeze some of the dough if you prefer (see step 9).
STORAGE: Store cookies in an airtight container at room temperature. They're best enjoyed the same day they're made (the marshmallows do harden every day a bit). Freeze cookie dough as opposed to baked cookies (see next step).
FREEZING DOUGH: Instead of freezing the baked cookies, freeze the dough! Drop the cookie dough balls on a large sheet pan and freeze until solid. Once solid, transfer the frozen cookie dough balls to an airtight container or bag and freeze for up to 3 months. To bake: Bake these cookies straight from the freezer. There is no need to thaw, but you may need to add a few extra minutes to the baking time. Bake until the edges are lightly browned, and the center is still soft.