Brown the butter in a medium saucepan over medium heat for 5-7 minutes until golden brown and nutty.
Mash the ripe bananas in a large mixing bowl and mix in the browned butter, brown sugar, and granulated sugar until well combined.
In another bowl, whisk together the gluten-free flour, baking soda, salt, and ground cinnamon.
Add eggs and vanilla extract to the banana mixture, then gradually mix in the dry ingredients until just combined.
Preheat the oven to 350°F. Pour the batter into a prepared 9x13-inch baking dish and bake for 25-30 minutes.
Let the bars cool in the pan for 10 minutes before transferring to a wire rack to cool completely.