Line a 9×13-inch baking dish with parchment paper, leaving some hanging over the sides for easy removal. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, or using a hand mixer, cream together the softened butter, peanut butter, granulated sugar, and brown sugar on medium speed until smooth and fluffy.
Add the egg and vanilla. Mix until fully combined.
In a separate bowl, whisk together the flour, baking soda, and salt.
Add the dry ingredients to the peanut butter mixture. Mix on low just until combined. Do not overmix.
Gently fold in most of the marshmallows, chocolate chips, and peanut butter chips, reserving some of each for the top.
Press the dough evenly into the prepared pan. Sprinkle the reserved marshmallows, chocolate chips, and peanut butter chips over the top, gently pressing them into the dough.
Cover and refrigerate for 1 hour, or until the dough is firm.
When ready to bake, preheat the oven to 350°F.
Bake for 18 to 22 minutes, or until the edges are lightly golden and the center looks set but still soft. Be careful not to overbake.
Let the cookies cool completely in the pan. They will continue to firm up as they cool.
Use the parchment paper to lift the cookies from the pan, then cut into squares.