Pre-heat oven to 350 degrees.
In a saucepan on medium heat, add butter and continuously stir until it has browned. Once there are white bubbles forming on top of the butter and the color is brown remove from heat immediately. Though the name is burnt butter and brown sugar banana bread, we don't actually want burnt butter :) .
Let the butter cool for at least 10 minutes. I like to transfer the butter to a ceramic bowl and place it in the freezer for a faster cooling process.
The butter needs to cool so it does not cook the raw eggs when mixed in.
In a large mixing bowl add three brown bananas (thawed if they are frozen) and mash them with a fork.
To the same bowl add: eggs, rum (optional), brown sugar, cooled browned butter and mix until combined.
Add the dry ingredients into the bowl: flour, baking soda, salt, and half of the walnuts. Stir until fully combined.
Add the flour slowly to control the consistency. If the batter is thickening fast then do not use the entire amount of flour. If the bread batter is too thin, add more flour.
Line a 9 x 5 loaf pan with parchment paper and spray with cooking oil then pour the mixture into the pan.
Sprinkle extra brown sugar and walnuts on top and bake for 40 minutes at 350 degrees.
Let cool for at least 5 minutes before enjoying.