In a saucepan over medium heat, melt the butter until it turns golden brown and has a nutty aroma. Remove from heat and let cool slightly.

In a mixing bowl, combine the mashed bananas, beaten egg, vanilla extract, and the browned butter. Mix well.

In another bowl, whisk together the flour, baking soda, salt, and brown sugar until well combined.

Gradually add the dry ingredients to the wet ingredients, stirring until just combined. Do not overmix.

Pour the batter into a greased loaf pan and smooth the top. Bake in a preheated oven at 350°F (175°C) for 60 minutes or until a toothpick inserted in the center comes out clean.

Allow the banana bread to cool in the pan for 10 minutes before transferring to a wire rack to cool completely. Slice and serve.