Preheat the oven to 375°F.
In a small saucepan, brown the butter. Do this by melting the butter, then letting it foam and cook for 3-5 minutes, stirring occasionally until flecks of brown form at the bottom of the pan. Transfer it to a bowl and whisk to let it cool slightly.
Whisk the sugar and butter together for a minute or so. Crack in the eggs and add the vanilla. Whisk vigorously into a smooth mixture.
Add the bananas directly into the bowl. Use a fork to mash them into small pieces, but you don't need the bananas completely smooth. Switch to your whisk and whisk together.
Add in the flour, baking powder and salt and stir to combine into a nice, thick batter.
Transfer the batter to a parchment-lined loaf tin and bake at 375°F for 50-55 minutes.