In a stainless steel skillet or a 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 about 5-10 minutes before proceeding.
Preheat the oven to 350°F (175°C). Line a 9×13-inch baking dish with parchment paper. Set aside.
In a large bowl, using a hand mixer on medium speed, beat the cooled browned butter and brown sugar until well combined and smooth. Add the vanilla extract and mix until incorporated.
Add the eggs, one at a time, mixing on low speed until fully combined.
Add the flour and salt, mixing on low until just incorporated. Do not overmix.
Gently fold in the mashed bananas until evenly distributed.
Pour the batter into the prepared baking dish and spread it into an even layer. Bake for 40–45 minutes, or until the center is set and a toothpick inserted in the middle comes out clean.
Let the bars cool completely before frosting with the buttercream..