In the bowl of a stand mixer fitted with the paddle attachment, combine the cold cubed butter, brown sugar and granulated sugar.
Mix on low for 2–3 minutes, until the butter is broken into small pieces and coated with sugar. Don't cream until light and fluffy. It should still look somewhat crumbly.
Add the wet ingredients
Add the egg, egg yolk and vanilla.
Mix on low just until incorporated.
Add the dry ingredients
Whisk together the flour, ½ cup crushed graham crackers, cornstarch, baking soda, baking powder and salt.
Gradually add to the mixer on low and mix until almost combined.
Fold in the chocolate chips, chopped chocolate and about ½ cup of the additional broken graham cracker pieces.
Don't overmix.
Stuff with marshmallows
Divide the dough into 12 portions.
Take one portion and divide it roughly in half. Flatten one half slightly and place one large marshmallow in the center. Place the remaining dough over it and gently seal the dough completely around the marshmallow.
Shape it into a tall, chunky mound, not a smooth round ball.
Repeat with remaining cookies.
Refrigerate the stuffed cookie dough for at least 2 hours.
Overnight is still best, but 2–3 hours should give you a nice thick cookie.
Keep the marshmallows completely enclosed while chilling. We'll expose them a little when they're almost finished baking.
Preheat oven to 370°F.
Line baking sheets with parchment and bake only 3–4 jumbo cookies per sheet.
Bake for about 9–10 minutes, then quickly pull the pan out.
Gently press a few pieces of graham cracker and extra chopped chocolate into the tops. You can also slightly open the top of the cookie to expose some marshmallow.
Return to the oven for another 2–4 minutes.
They're done when the edges are lightly golden, the tops are set, and the centers still look a little underbaked.