Preheat oven to 350 degrees. Spray a 9 x 13 baking dish with cooking spray and line with parchment. Set aside.
In a large bowl, beat together the butter, peanut butter, and sugars, until light and fluffy, about 2 minutes. Add in the eggs, banana, and vanilla and mix until combined. Add in the flour and salt and mix to combine. Fold in the peanuts and chocolate chips.
Pour the batter into the prepared baking dish using a spatula or damp hands to spread evenly in the pan. Bake for 35-40 minutes, or until a toothpick inserted in the bars comes out clean. (Note that this dish will not brown much so you will have to check for doneness using a toothpick.) Cool in pan for 10 minutes, then remove bars using parchment to cool on a wire rack completely before cutting.
Notes
This is a great make-ahead recipe that can be refrigerated for up to a week.