To brown butter, heat unsalted butter in a light color saucepan over medium heat. Once the butter melts, it will become foamy. When the foam subsides, you will see clear bubbles, start whisking continuously at this point to keep the milk solids from burning. When the butter is caramel in color and smells nutty, turn off the heat and set aside to cool. This will take approximately 8 minutes and 30 seconds.
To prepare the banana, remove peels and microwave the bananas in a large bowl uncovered for 5 minutes. Transfer the cooked bananas to a strainer set over another bowl to extract the liquid. Put the banana pulp into a large mixing bowl and set aside to cool.
Transfer the banana liquid into a saucepan. Cook the liquid over medium heat until reduced by half and slightly caramelized. Set aside to cool.
Preheat the oven to 350°F.
Brush a loaf pan with some melted butter and line the bottom with a piece of parchment paper, letting the excess hang over the sides. Brush the parchment paper with more butter. Set aside.
When the brown butter, banana, and banana liquid cool to room temperature, mash the banana with a potato masher and add the banana liquid and brown butter to it. Whisk together to combine.
Add brown sugar, eggs, vanilla, and beat until smooth.
Sift flour, baking soda, and baking powder into a small mixing bowl. Stir in the kosher salt. Add the flour mixture to the liquid mixture and fold until just combined.
Transfer the batter to the prepared loaf pan and bake for 55 minutes until a toothpick inserted in the center of the loaf comes out clean with just a few crumbs attached.
Allow the bread to cool in the pan for 10 minutes before removing it from the pan and let it cool further on a wire rack.