Brown the butter in a light skillet over medium heat until nutty and amber; cool 5 minutes.

Preheat oven to 350°F (175°C) and line baking sheets with parchment.

Whisk browned butter with both sugars; stir in mashed banana, egg, and vanilla.

Fold in flour, baking soda, and salt until just combined; stir in toffee bits.

Drop 2-tablespoon scoops onto sheets and bake 10–12 minutes until edges set.

Cool 5 minutes on the sheet, then transfer to a rack to finish cooling.