In a medium saucepan over medium heat, melt the butter. Continue cooking, stirring frequently, until the butter turns a golden brown color and develops a nutty aroma, about 5-7 minutes. Be careful not to burn it.

In a large mixing bowl, combine the mashed bananas, egg, brown sugar, granulated sugar, and vanilla extract. Mix well until fully combined.

Pour the brown butter into the banana mixture, making sure to scrape all the browned bits from the bottom of the pan. Stir until well incorporated.

In a separate bowl, whisk together the baking soda, salt, and flour. This ensures even distribution of the leavening agent.

Gently fold the dry ingredients into the wet mixture using a rubber spatula. Mix until just combined; do not overmix. If using, fold in the chopped nuts.

Preheat your oven to 350°F (175°C). Grease the loaf pan with butter or line it with parchment paper for easier removal.

Pour the batter into the prepared loaf pan and smooth the top with the spatula.

Bake in the preheated oven for 55-65 minutes, or until a toothpick inserted into the center comes out clean. If the top starts to brown too quickly, cover it loosely with aluminum foil.

Once baked, remove the banana bread from the oven and let it cool in the pan for 10 minutes. Then, transfer it to a wire rack to cool completely before slicing.