In a medium saucepan, melt the unsalted butter over medium heat, stirring frequently until it turns golden brown, about 4-5 minutes. Remove from heat and let cool slightly.

Preheat your oven to 340°F (170°C). Grease a 9x9-inch baking pan and line it with parchment paper.

In a large mixing bowl, combine the warm browned butter with light brown sugar and mix until smooth and glossy, about 1-2 minutes.

Beat in the eggs one at a time, stirring until the batter becomes pale and thick, about 2-3 minutes.

Mash the ripe bananas in a separate bowl and mix them into the batter along with vanilla extract.

Sift together the flour, baking powder, and salt in another bowl. Gradually fold this into the banana mixture until combined.

Stir in dark chocolate chips and chopped walnuts until evenly distributed.

Pour the batter into the prepared baking pan and bake for 30 minutes, or until a toothpick comes out almost clean.

Remove from oven and let cool on a wire rack for 20 minutes. Use parchment to lift out and cool completely before cutting.