Preheat the oven to 350°F and prepare an 8x8-inch baking pan with parchment paper or lightly grease the pan.
Mash the very ripe bananas until mostly smooth.
In a large mixing bowl, combine the mashed bananas, granulated sugar, egg, and vanilla extract
Add the all-purpose flour, baking powder, and salt.
Stir gently until the ingredients are just combined. Do not overmix.
Fold in the chocolate chips, if using.
Spread the batter evenly into the prepared baking pan.
Bake for 25–30 minutes, or until the center is set and a toothpick inserted into the center comes out with a few moist crumbs.
Remove from the oven and allow the bars to cool completely.
To make the brown butter frosting, melt the butter in a saucepan over medium heat.
Continue cooking until the butter develops a golden-brown color and nutty aroma, watching carefully so it does not burn.
Remove the butter from the heat and allow it to cool slightly.
Add the confectioners' sugar, vanilla, milk, and salt.
Stir until the frosting is smooth and spreadable. Add a small amount of additional milk if needed to reach the desired consistency.
Spread the brown butter frosting over the completely cooled banana bars.
Allow the frosting to set slightly before slicing into squares.