Preheat oven to 350ºF (180ºC). Line baking sheets with parchment paper.
Beat butter and sugars in a large mixing bowl, using an electric mixer on medium speed, until blended, about 1 minute. Add egg and vanilla and mix until blended.
Combine flours, oats, baking soda and salt in medium mixing bowl. On low speed, add the flour mixture and mix until incorporated. Mix in chocolate chips. Add quartered bananas and mix on low just until they are evenly distributed and some of the banana is mashed but most of the pieces are still visible.
Drop 2 tbsp (30mL) dough on prepared baking sheets, about 2” (5cm) apart. Place banana chip on top of each cookie.
Bake in preheated oven for 20 minutes until the edges are lightly browned. Cool cookies on baking sheets for 5 minutes. Transfer cookies to cooling rack.