Position a rack in the center of the oven and heat the oven to 350°F. Line a 9-inch-square metal baking pan with foil, leaving an overhang on two sides for easy removal of the brownies. Butter and flour the bottom and sides of the foil, tapping out the excess flour.Melt the butter in a 3-quart saucepan over medium heat until it smells nutty and turns golden, 4 to 5 minutes. Remove the pan from the heat and let cool for 5 minutes. Whisk in the sugar and salt, followed by the eggs, banana, and vanilla. Whisk in the cocoa powder and flour, mixing slowly at first and then more vigorously until the batter is combined.
Spread the batter in the prepared baking pan, smoothing it so it fills the pan evenly. 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, 40 to 45 minutes. Let the brownies cool in the pan before topping.
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