• In a stainless steel skillet or light-bottomed pan over medium-high heat, melt the butter, swirling the pan occasionally. Once the butter begins to foam and brown specks form at the bottom, stir continuously to prevent burning.
• When the butter turns a deep golden brown and smells nutty, immediately remove it from the heat and pour it into a large heat-safe bowl. Let it cool for 5 to 10 minutes.
• Preheat oven to 350°F (175°C).
• Line a 9×13-inch baking dish with parchment paper, leaving an overhang for easy removal. Set aside.
• Using a hand mixer on medium speed, beat the cooled browned butter and brown sugar in the large bowl until well combined and smooth.
• Add the vanilla extract and mix until fully incorporated.
• Add the eggs, one at a time, mixing on low speed until combined.
• Add the flour and kosher salt, mixing on low speed just until no dry streaks remain. Do not overmix.
• Gently fold in the mashed bananas until evenly distributed throughout the batter.
• Pour the batter into the prepared baking dish and spread into an even layer with a spatula.
• Bake for 32 to 36 minutes, or until the center is set and a toothpick inserted into the middle comes out with a few moist crumbs but no wet batter.
• Let the bars cool completely in the pan on a wire rack before frosting.

Browned Butter Frosting

• To make the frosting, melt the butter in a small saucepan over medium heat. Continue cooking, swirling the pan, until the butter turns golden brown and smells nutty, about 3 to 5 minutes. Remove from the heat and let it cool slightly.
• Transfer the browned butter to a medium mixing bowl. Add the confectioners’ sugar, whole milk, vanilla extract, and a pinch of kosher salt.
• Using a hand mixer, beat the mixture until smooth and creamy. Add more milk if needed to reach a spreadable consistency.
• Once the banana bars have cooled completely, spread the browned butter frosting evenly over the top.
• Cut into bars and serve.