Cut the cold butter into small pieces and add the butter to the bowl of a stand mixer. Add the white sugar and brown sugar to the bowl as well, and beat until light and fluffy.
In a separate mixing bowl, sift or whisk together the all-purpose flour, cake flour, baking soda, and salt. Add the dry ingredients to the the wet ingredients and mix together.
Form a ball of cookie dough. Make an indentation in the cookie dough ball and place 1/2 of a jumbo marshmallow in the center. Fold the dough around it while forming it back into a ball.
Refrigerate the cookie dough balls for 30 minutes. Preheat your oven to 375 degrees. Space the cookie dough balls on a parchment-lined baking sheet about 2 inches apart.
Bake for 8 minutes, then place one Hershey bars piece on top of the cookie and bake for 2 more minutes. Allow to rest, then transfer to a wire rack. Enjoy!