Add butter to a light-colored saucepan and cook over medium-low to medium heat, stirring frequently, until melted and browned (see notes for more details). Allow to cool to room temperature.
Add the browned butter, light brown sugar, and sugar in a bowl. Mix until well combined.
Add the egg yolk, vanilla extract, salt, and baking soda and mix until thoroughly combined, making sure to scrape down the bottom and sides of the bowl.
Add the mashed banana and mix until combined.
Add the flour and mix until just combined.
Fold in the chocolate chips.
Scoop dough into 1.5 tbsp mounds (#40 cookie scoop) and place on a prepared baking sheet.
Bake for 8-10 minutes or until edges are golden brown and the tops look set.
Allow to cool on the baking sheet for 1-2 minutes before transferring to a wire rack to cool completely.
Notes
Brown Butter - While cooking, the butter will begin to bubble and foam. After several minutes, things will quiet down and the milk solids will begin to toast. At this point, stir constantly until the milk solids are golden brown and the butter smells nutty. Transfer to a heat-safe container and allow to cool to room temperature.For more information, see my blog post on How to Brown Butter (https://brooksidekitchen.com/how-to-brown-butter/)