Preheat oven to 350°F Line a baking sheet with parchment paper.
In a medium bowl, whisk together flour, baking powder, baking soda and salt.
In a large bowl, cream together sugars and butter. Beat in egg until smooth, then mix in the peanut butter and mashed banana. Working by hand or with a mixer on low speed, blend dry ingredients into the peanut butter mixture. Stir in the oats.
Shape dough into tablespoons and place on baking sheet, flattening each slightly with your fingers.
Bake for 10-12 minutes, until cookies just begin to turn golden around the edges.