In a stand mixer or with a hand mixer, add your butter and sugars and cream together until fluffy, about 3 - 5 minutes. Add your egg and vanilla. Scrape down the sides of the mixer as needed.
In a food processor, add your graham crackers and process until they are the consistency of flour, about 3 minutes. Pour into a bowl and whisk together with all-purpose flour, corn starch, baking soda and salt.
Add flour mixture to butter until combined. Stir in chocolate chips and marshmallow pieces. Chill dough in refrigerator for 30 - 60 minutes.
Preheat the oven to 350 and line two baking sheets with parchment.
Roll heaping tablespoons of dough in your hands to create about 30 balls. Place balls on prepared baking sheets and bake in the oven for 8 - 10 minutes or until cookies edges are brown and crispy but center is soft. Repeat with remaing balls.
Allow cookies to cool and then enjoy