Preheat the oven to 350° Fahrenheit.
Slice the butter up into 8 pieces. Then add the sliced butter to a medium sauce pan, over medium heat.
Let the butter melt, and then foam. Stirring occasionally, to keep the butter from bubbling over. The butter will begin to form brown flakes, keep stirring it, and all of it will brown. Remove the butter from the heat and transfer it to a bowl to cool. Set aside.
In a mixing bowl, stir together the flour, baking soda, and salt. Set aside.
Remove the overripe bananas from the peel and add the bananas to a mixing bowl. Use a fork to mash them up so there are very few lumps. Whisk in the egg, and vanilla extract. Then stir in the brown sugar, and slowly add in the brown butter. Stir together the wet ingredients until they're all mixed together.
Fold the dry ingredients into the wet ingredients, until they are just combined.
Line a large loaf tin with parchment paper, then pour the banana bread batter into the loaf tin.
Bake the banana bread for 60 minutes, or until a toothpick can be inserted into the middle and comes out clean.
Remove the banana bread from the oven, and let cool for 15 minutes, use a knife to gently loosen the banana bread from the sides of the pan. Carefully flip the pan over and remove the banana bread from the pan. Remove the parchment paper from banana bread, and then let the banana bread cool on a cooling rack. Store in an airtight container for up to a week, for optimal freshness.