Make Brown Butter: Place stick of butter into a small skillet or saucepan over medium-low heat. Swirl constantly for 2-3 minutes or until it is light brown in color and has a nutty aroma. Set aside and let cool.
Once the butter is cooled, preheat oven to 350°. Combine melted brown butter and both sugars in a medium bowl and stir by hand until incorporated. Add egg, vanilla and mashed bananas. Stir until mixed well. Add flour, oats, baking soda, cinnamon and salt to wet ingredients and mix until everything is completely incorporated.
Using a medium cookie scoop, scoop mounds of dough onto a silicone mat {or parchment} lined baking sheet. Using your fingers, slightly press down each cookie. Bake for 10-11 minutes or until the edges of the cookies begin to turn golden brown and the middle is set. Remove from the oven and let them cool on the cookie sheet for 10 minutes. Continue to bake until all of the dough is gone.
Store in an air tight container at room temperature for up to one week or freeze for up to 3 months.