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

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

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

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 55-60 minutes, or until a toothpick inserted in the center comes out clean.

Allow the banana bread to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely before slicing.