To Make the Banana Blondies:
Prep: Preheat your oven to 350°F (175°C). Line a 9x9 inch baking pan with parchment paper, leaving an overhang on two sides to act as handles for easy removal later.
Mix Wet Ingredients: In a large bowl, whisk together the melted butter and packed brown sugar until well combined. Add the eggs and vanilla extract and whisk until the mixture is smooth and glossy.
Combine: Add the flour and salt to the wet ingredients, then add the mashed banana. Using a rubber spatula or mixing on low speed, fold the ingredients together just until the last streaks of flour disappear. Do not overmix. A few lumps are okay.
Bake: Pour the batter into the prepared pan and spread it into an even layer. Bake for 25-28 minutes, or until the edges are golden brown and a toothpick inserted into the center comes out with moist, sticky crumbs attached (not wet batter).
Cool Completely: Place the pan on a wire rack and let the blondies cool completely to room temperature. This step is crucial, otherwise the frosting will melt and slide off.
To Make the Brown Butter Frosting:
Brown the Butter: In a small saucepan over medium-low heat, melt the ¼ cup of butter. Continue to cook, swirling the pan frequently, until the butter foams up and you see nutty-smelling brown specks forming on the bottom. Watch carefully as it can go from brown to burnt very quickly.
Make the Glaze: Once the butter is browned, immediately add the brown sugar and milk. Stir constantly until the mixture comes to a gentle boil.
Cool Slightly: Remove the pan from the heat and let it cool for 5-10 minutes. This prevents the frosting from becoming too thin.
Finish the Frosting: Whisk the powdered sugar into the brown butter mixture until it's completely smooth.
Frost and Set: Pour the warm frosting over the completely cooled blondies and spread it into an even layer. Let the frosting set for at least 30 minutes before using the parchment handles to lift the blondies out of the pan and cut them into squares.