Preheat oven to 350°. Lightly coat an 8x8” baking dish with butter and line with parchment paper. Whisk oats, flour, baking powder, cinnamon, and salt in a large bowl; set aside. Whisk brown sugar, eggs, and vanilla in another large bowl until mixture is smooth.
Slowly stream butter into brown sugar mixture, whisking constantly until well blended. Add bananas to dry ingredients; toss to coat. Stir into brown sugar mixture (batter will be thick). Spread batter in prepared dish. Bake until cake is golden brown and the edges start to pull away from sides of dish, 35–40 minutes. Transfer to a wire rack and let cool completely in dish. Cut into 12 servings.