Place butter, granulated sugar, and brown sugar in the bowl of a large stand mixer, fitted with a paddle attachment OR in a large mixing bowl with a wooden spoon. Mix on medium speed or by hand until creamy, about 2 minutes. Add egg and vanilla, beating well until smooth.
In a separate medium bowl, combine flour, graham crackers, baking soda, and salt. Fold into butter mixture until dough is just combined, then fold in chocolate chips.
Scrape dough into a ball and wrap in plastic wrap. Refrigerate dough 2 hours or up to 6 days.
When ready to bake cookies, preheat oven to 350F. Line 2 large sheet pans with parchment paper and set aside.
Remove dough from fridge and use a cookie scoop to scoop dough out into 1-in balls. Place balls 2 inches apart on prepared sheet pans and top with additional graham cracker pieces and chocolate chips.
Bake cookies at 350F 10-12 minutes until edges of cookies are golden brown. Cool cookies 10 minutes on sheet pan then transfer to a cooling rack to cool completely. Enjoy!