Preheat your oven to 350°F (175°C) and line your 9x9-inch baking pan with parchment paper.
In a small saucepan, melt 1 cup of unsalted butter over medium heat. Stir occasionally until it turns golden brown and smells nutty, about 5-7 minutes.
In a large mixing bowl, combine the melted brown butter and 1 cup of packed brown sugar. Whisk until well combined, then add in the room temperature egg and 1 teaspoon of vanilla extract. Mix until smooth.
In another bowl, whisk together 1 cup of all-purpose flour, 1 teaspoon of baking powder, and 1/2 teaspoon of salt.
Fold the dry ingredients into the wet mixture until just combined. Then, fold in 1 cup of mashed banana and 1 cup of chopped walnuts.
Pour the batter into the prepared baking pan and spread it evenly. Bake for 25-30 minutes, or until a toothpick inserted in the center comes out clean.
While the blondies cool, prepare the glaze by melting 1/2 cup of unsalted butter in a small saucepan. Whisk in 1 cup of powdered sugar until smooth.
Drizzle the brown butter glaze over the cooled blondies and let it set for a few minutes before cutting them into squares.