Preheat your oven to 350°F (175°C). Grease and line a 9x13-inch baking pan with parchment paper, leaving some overhang for easy removal.

In a large mixing bowl, combine the mashed bananas, granulated sugar, brown sugar, and melted butter. Mix until well combined.

Add the eggs and vanilla extract to the banana mixture, stirring until smooth.

In a separate bowl, whisk together the baking soda, salt, and flour. Gradually add the dry ingredients to the wet ingredients, mixing just until combined. If using, fold in the chopped nuts.

Pour the batter into the prepared baking pan, spreading it evenly. Bake for 20-25 minutes, or until a toothpick inserted in the center comes out clean.

While the bars are baking, prepare the brown butter frosting. In a small saucepan, melt the butter over medium heat. Continue cooking, stirring frequently, until the butter turns a golden brown color and has a nutty aroma. Remove from heat and let it cool slightly.

In a mixing bowl, combine the browned butter, powdered sugar, vanilla extract, and 2 tablespoons of milk. Whisk until smooth. If the frosting is too thick, add more milk, one tablespoon at a time, until desired consistency is reached.

Once the bars are cool, spread the brown butter frosting evenly over the top. Cut into squares and serve.