Brown the butter in a light-colored medium saucepan or pot, place butter inside and cook on low medium heat. No need to stir consistently, just allow the butter to melt and begin cooking.
About halfway through the butter will begin to foam up a bit, but you only need to take your rubber spatula and stir it to bring it back down a bit. Continue to cook on the same heat, and about 6 to 7 minutes in you will begin to see light foaming on the top.
You will notice along the edges browning, and if you stir the butter there will be small brown bits at the bottom of the pan (milk solids). Begin stirring at this point until the color is a light to medium color and you smell the nutty aroma of the butter.
Remove from heat and pour into a heat safe medium bowl. Allow to cool.
Preheat oven to 350 F. Combine all-purpose flour, baking soda, and fine sea salt in a separate bowl and whisk until well combined.
Add sugars and vanilla extract to browned butter, and whisk until well combined.
Add eggs one at a time and mix well before adding next one.
Add mashed bananas to mixture and mix until well combined.
Add dry ingredients and fold into wet ingredients with rubber spatula.
Fold in chocolate chips with rubber spatula until just combined.
Line 9x5 loaf pan with parchment paper or baking spray. Pour batter into loaf pan. Top with chocolate chips if desired.
Bake for 55 - 60 minutes, or until a toothpick comes out with only a few moist crumbs.
Remove from heat and allow to cool completely before slicing.