Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium saucepan over medium heat, melt the butter. Continue cooking, stirring frequently, until the butter turns golden brown and has a nutty aroma, about 5-7 minutes. Remove from heat and let cool slightly.
In a large mixing bowl, combine the brown sugar and granulated sugar. Add the browned butter and mix until well combined.
Beat in the eggs one at a time, followed by the vanilla extract and mashed bananas.
In another bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Fold in the oats and chocolate chips.
Scoop tablespoon-sized portions of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 10-12 minutes, or until the edges are golden brown.
Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.