Brown the butter: In a saucepan over medium-low heat, melt the butter. Continue to cook, stirring occasionally, until the butter turns a golden brown color and has a nutty aroma. Remove from heat and allow to cool slightly.

Mix sugars and butter: In a large mixing bowl, combine the brown sugar, granulated sugar, and slightly cooled brown butter. Mix until well combined.

Add eggs and vanilla: Add the egg, egg yolks, and vanilla extract to the sugar-butter mixture. Mix until smooth and creamy.

Combine Dry Ingredients: In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.

Combine Wet and Dry Mixtures: Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Fold in the semi-sweet chocolate chips.

Chill the Dough: Cover the cookie dough with plastic wrap and refrigerate for at least 30 minutes, or up to 24 hours, to allow the flavors to meld and the dough to firm up.

Preheat the Oven: Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.

Prepare the Graham Crackers: Break a graham cracker sheet in half to create two squares. Place them on a baking sheet lined with parchment paper or a silicone mat.

Add Chocolate Squares: Place two Hershey's chocolate squares on top of each graham cracker square.

Prepare Marshmallows: Cut a marshmallow in half horizontally using kitchen shears or scissors.

Assemble the Cookies: Place the sticky side of the marshmallow halves on top of the chocolate squares.

Shape the Cookie Dough: Take a portion of the chilled cookie dough and flatten it in the palm of your hand to form a disc.

Encase the Filling: Place the flattened cookie dough on top of the marshmallow-topped graham cracker. Seal the edges of the cookie dough around the filling, ensuring it is completely covered.

Bake the Cookies: Place the assembled cookies on the prepared baking sheet, leaving space between each cookie to allow for spreading. Bake in the preheated oven for 12-14 minutes, or until the edges are lightly golden.

Cool and Enjoy: Remove the cookies from the oven and allow them to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely. Enjoy these gooey, indulgent s'mores stuffed chocolate chip cookies with a glass of cold milk or a scoop of vanilla ice cream.