Preheat oven to 350°F (175°C). Grease and flour a loaf pan.
In a bowl, whisk together flour, baking soda, and salt.
In a separate bowl, whisk together browned butter, granulated sugar, and brown sugar until combined. Add eggs one at a time, then stir in mashed bananas and vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Pour batter into the prepared loaf pan and bake for 50-60 minutes, or until a toothpick inserted into the center comes out clean.
Let the bread cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.