Preheat the oven to 350°F (175°C). Grease the loaf pan or line it with parchment paper.
In a medium saucepan, melt the butter over medium heat. Continue cooking until the butter turns a golden brown color and has a nutty aroma, about 5-7 minutes. Remove from heat and let cool slightly.
In a mixing bowl, combine the mashed bananas and granulated sugar. Stir well.
Whisk in the browned butter, making sure to scrape up any bits from the bottom of the pan.
Add the egg and vanilla extract to the banana mixture and mix until smooth.
In another bowl, combine the baking soda, salt, cinnamon, and flour. Stir until well mixed.
Gradually fold the dry ingredients into the wet ingredients just until combined. Do not overmix.
Pour the batter into the prepared loaf pan, smoothing the top if needed.
Bake for 60 minutes, or until a toothpick inserted in the center comes out clean.
Allow the bread to cool in the pan for 10 minutes before transferring to a wire rack to cool completely before slicing.