Preheat oven to 350°F. Line an 8x8-inch pan with parchment paper or foil and lightly spray the liner with non-stick cooking spray.
In a large bowl, whisk together ½ cup melted unsalted butter, ¾ cup packed brown sugar, 1 room temperature egg, and 1 teaspoon vanilla until smooth.
Add 1 cup all-purpose flour, ½ teaspoon baking powder, and ¼ teaspoon salt to the bowl and stir until just incorporated.
Fold in ½ cup mashed banana (about 1 medium banana) and 1 cup chopped walnuts until evenly combined.
Pour the batter into the prepared pan and spread it into an even layer. Bake for 20–25 minutes, or until the top is lightly brown and a toothpick inserted in the center comes out clean or with a few moist crumbs.
Remove the pan from the oven and let the bars cool in the pan until completely cool to the touch (cooling to room temperature is best) before glazing.
To make the brown butter glaze, melt ¼ cup unsalted butter in a medium saucepan over medium heat. Continue to cook, whisking or swirling, until the butter turns golden brown and gives off a nutty aroma. Remove from heat immediately to avoid burning.
Whisk ¾ cup powdered sugar into the browned butter until the mixture is smooth and evenly combined.
Pour the glaze over the cooled bars and spread evenly with a spatula. Allow the glaze to set before cutting into bars and serving.