Preheat the oven to 350 F and grease and parchment line (with overhangs so it's easy to remove) an 8x8 inch square baking pan.
In a food processor, pulse ¾ cups of the rolled oats until they become smaller. Sand texture is too fine, but you definitely want to break them down a bit.
Combine the 1 ¾ cups flour, ground oats, baking soda and salt in a bowl and set aside.
In a stand mixer fitted with the paddle attachment (or with a handheld mixer), beat the butter, peanut butter, and sugar until combined. Add the egg and vanilla, mix until incorporated, then scrape down the sides of the bowl. Add the dry ingredients and mix until just combined.
Add ⅔ of the mixture to the prepared pan and press down evenly into all corners and sides.
In a separate bowl, combine the leftover ⅓ of the flour and oat mixture, add in the remaining ¼ of whole rolled oats and bring together with a fork. You will have to break up the mixture a bit with the fork, but you want to keep the pieces crumbly (think a streusel topping).
Next, prepare your bananas. Slice two bananas into ¼ inch thick rounds and mash the other two bananas with a fork. Add the 1.5 T of flour to the mashed bananas and mix to combine.
Spread the mashed bananas evenly over the base of the bars, then layer the banana slices on top. Sprinkle the remaining oat crumbles evenly over top of the bars.
Bake for 30 - 35 minutes until the edges are slightly browned.
Let cool completely in the pan, then remove and cut into 9 or 16 pieces.
Store the bars covered at room temperature for 2 days.