Preheat your oven to 350°F (175°C). Grease a 9x5-inch loaf pan or line it with parchment paper.
In a medium saucepan, melt the butter over medium heat. Cook the butter, stirring frequently until it turns golden brown and develops a nutty aroma, about 5-7 minutes. Be careful not to burn it. Once browned, remove from heat and let it cool slightly.
In a large mixing bowl, combine the mashed bananas with the brown sugar and granulated sugar. Mix well until combined.
Add the eggs and vanilla extract to the banana mixture, whisk until smooth.
Pour the slightly cooled brown butter into the banana mixture and stir until well incorporated.
In another bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg. Gradually fold the dry ingredients into the wet mixture until just combined. Avoid overmixing.
If using, gently fold in the chopped walnuts or pecans.
Pour the batter into the prepared loaf pan and smooth the top.
Bake in the preheated oven for 60-65 minutes or until a toothpick inserted in the center comes out clean.
Remove from the oven and let the banana bread cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.
Notes
Slice the banana bread and serve warm with a pat of butter or a drizzle of honey on top. You can also garnish with a sprinkle of cinnamon for added flair.