Step 1: Preheat the Oven

1. Preheat your oven to 350°F (175°C).
2. Grease and line a 9×9-inch baking pan with parchment paper, making sure the paper hangs over the edges for easy removal later.

Step 2: Brown the Butter

1. In a small saucepan, melt the 1/2 cup of unsalted butter over medium heat. Continue cooking until it turns a golden brown color and gives off a nutty aroma. Be sure to stir frequently to prevent burning.
2. Remove from heat and let it cool slightly.

Step 3: Mix the Wet Ingredients

1. In a large bowl, whisk together the granulated sugar, brown sugar, and the browned butter.
2. Add in the eggs one at a time, beating well after each addition.
3. Stir in the vanilla extract and mashed bananas until well combined.

Step 4: Combine the Dry Ingredients

1. In a separate bowl, whisk together the flour, baking soda, and salt.
2. Gradually add the dry ingredients to the wet mixture, stirring until just combined. Do not overmix.
3. If desired, fold in the chopped walnuts and chocolate chips.

Step 5: Bake the Brownies

1. Pour the batter into the prepared baking pan and spread it out evenly.
2. Bake in the preheated oven for 25-30 minutes or until a toothpick inserted into the center comes out with a few moist crumbs.
3. Remove from the oven and allow the brownies to cool in the pan for about 10 minutes. Then, using the parchment paper overhang, lift them out of the pan and transfer to a wire rack to cool completely.

Step 6: Make the Frosting (Optional)

1. In the same small saucepan you used earlier, brown the 1/4 cup of unsalted butter.
2. In a medium bowl, whisk together the powdered sugar, browned butter, vanilla extract, and milk. Adjust the consistency by adding more milk if necessary. The frosting should be thick but spreadable.
3. Once the brownies are fully cooled, spread the frosting evenly over the top.

Step 7: Slice and Serve

1. After frosting, slice the banana bread brownies into squares and serve.