Start by preparing the browned butter by gently heating the butter in a saucepan over low heat. Stir regularly until it takes on a brown color and develops a nutty aroma. Be careful, as this can happen quickly, and remove it from heat as soon as it begins to brown. Set aside to cool.
Mash the bananas with a fork in a mixing bowl.
Add the eggs, both sugars, baking soda, and salt and mix thoroughly again.
Add the cooled browned butter, mix again, and finally add the flour.
After adding the flour, do not overmix to keep the cake moist: mix gently and briefly until just homogeneous.
Notes
Ensure not to overmix the batter to maintain the texture.