Preheat your oven to 350 degrees F. Spray your bread pan with cooking spray or grease it with butter. Make sure you grease the pan well so the bread doesn't stick!
In a small skillet pan melt your butter. Keep it on a low/medium heat and let it start bubbling. Stir the butter consistantly to keep it from burning. Once you see little brown flecks appear and can smell the brown butter remove it from the heat. Brown butter can burn very easily so be careful not to keep it on the heat for too long. Let the brown butter cool for a 5-10 minutes.
Mash your bananas on a plate or in a medium sized bowl. Make sure to mash the bananas well.
In the medium sized bowl, add the bananas, brown butter and light brown sugar. Whisk these ingredients together until they are combined.
Add the egg and vanilla to the banana mixture and mix well.
Add the cinnamon, nutmeg, baking soda and salt to this banana mixture. Whisk until combined.
Add your flour 1/2 cup at a time to this mixture and stir with a spoon. Mix together until well combined, but don't overmix.
Pour the batter into your prepared bread pan. Bake for 45 minutes or until a toothpick inserted in the middle comes out clean.
Let the bread cool slightly before removing it from the pan and allowing to cool on a wire rack.