Preheat oven to 375º and line two large baking sheets with parchment paper. Make cookie dough: In a large bowl using an electric mixer (or in a stand mixer), beat butter and sugars until light and fluffy, about 2 minutes. Slowly beat in vanilla extract and eggs until evenly incorporated.
In a separate bowl, mix flour, baking soda, and salt. Stir into butter-sugar mixture until just combined, then fold in chocolate chips.
Make s'mores: Cut each marshmallow in half lengthwise. Sandwich halved marshmallow and one square chocolate between two squares of graham cracker. Repeat with remaining s'more ingredients.
Assemble cookies: Scoop approximately 1/4 cup of cookie dough, flatten into a round slightly larger than the s'more, place on prepared baking sheet, and place prepared s'more on stop. Scoop another 1/4 cup of cookie dough, flatten, and use to cover s'more until no graham cracker is visible. Flip cookie so first scoop of dough is now the top of the cookie. Repeat until remaining dough and s'mores are used. Refrigerate on baking sheets for 10 to 15 minutes.
Bake cookies until lightly golden, 16 to 18 minutes. Let cool 10 minutes on baking sheet before serving.