Position a rack in the center of the oven and heat the oven to 350°F. Line an 8-inch-square metal baking pan two pieces of crisscrossing parchment paper, leaving a slight overhang on two sides for easy removal of the brownies. Set aside.
In a large saucepan over medium heat, melt the butter completely. Remove the pan from the heat and whisk in the brown and granulated sugars and salt until combined and thickened. Add the eggs one at a time, mixing well after each addition. Add the mashed banana and vanilla extract and whisk until combined. Add the cocoa powder and whisk until combined and no lumps remain. Fold in the flour until combined. Be careful not to over mix.
Spread the batter in the prepared baking pan, smoothing it so it fills the pan evenly. Place in the oven and bake until a toothpick or a skewer inserted in the center of the pan comes out with just a few moist clumps clinging to it, 30 minutes, but no longer than 35. Allow the brownies to cool in the pan while you make the topping.