Preheat oven to 350 degrees F. Grease an 8-by-8-inch square baking pan nonstick cooking spray and set aside.
First, brown the butter. Cut the butter into tablespoons. Place in a medium skillet. Melt the butter over medium heat, swirling it in the pan occasionally. The butter will foam and pop, so be careful. Continue to swirl the pan often. Remove pan from heat once the butter starts to brown and smells nutty. There will be small brown bits on the bottom. The butter should be an amber color.
In a large bowl, combine the browned butter and brown sugar. Stir until smooth. Add in the egg and stir until combined. Stir in the mashed banana and vanilla extract and stir until smooth.
Add the baking powder, baking soda, sea salt, and flour. Stir until just combined. Don't over mix. Fold in the chocolate chips.
Pour batter into prepared baking pan, spreading into an even layer. Bake for 30 to 35 minutes or until center is set and edges are light golden brown, and a toothpick inserted near the center comes out clean. Place pan on a wire rack and let cool completely before cutting into squares.