Heat oven to 350 degrees. Prepare a cookie sheet (about 12" x 18") and spray with cooking spray.
In a large bowl with a hand mixer, or in the bowl of a stand mixer with the paddle attachment, combine the butter, peanut butter, sugar, and brown sugar. Beat together for 1-2 minutes until it's pale in color and fluffy looking.
Add in the vanilla extract and eggs. Beat together until combined.
Add the quick oats, old-fashioned oats, and baking soda. Mix on low speed until combined. Add chocolate chips and butterscotch chips (reserve some for top of bars if wanted) and mix together just until combined.
Dump the cookie dough onto the prepared cookie sheet and spread out evenly. It will be sticky so use the back of a silver table spoon, or the butter wrapper, or spray your hands lightly with cooking spray.
Sprinkle the reserved chocolate chips and butterscotch chips on top and gently press them into the bars.
Cook for 17-20 minutes. I have never cooked these bars for longer than 17 minutes. Because of the no flour, they will appear underdone and slightly gooey but that's normal. Take them out of the oven and place the cookie sheet on a cooling rack. Let the bars cool for 30-45 minutes. They will come together and finish cooking slightly while cooling.
Use a pizza wheel cutter to cut these bars easily. Leftovers can be stored at room temperature covered. Or, place the bars in sandwich size Ziploc bags and store in the freezer. Eat cold, let sit at room temperature, or microwave for 10 seconds or so.