Preheat the oven to 350 degrees F. Line an 8 x 8 inch baking pan with parchment paper.

Place the butter in a saucepan and heat on medium until the butter is completely melted. Turn down the heat to medium-low and whisk the butter. Continue to cook while whisking until the butter begins to brown and smell nutty. Set aside to cool.

Beat the butter and both sugars together until lighter in color. Beat in the egg and vanilla until well combined.

Mix together the flour, baking soda, and salt in a separate bowl. Set aside. Mix together the banana and sour cream in another bowl. Set aside.

Alternate between mixing the flour mixture and the banana mixture into the wet ingredients. Begin and end with the flour.

Pour the batter into your baking pan and bake for 25 - 30 minutes, or until a toothpick inserted in the middle comes out clean or with a few crumbs sticking to it. Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.

Make the brown butter frosting. Brown the butter in the same manner you did for the banana cake. Place the browned butter in your mixing bowl and place in the refrigerator to firm slightly - about 10 minutes. Remove the butter from the refrigerator which is is mostly solidified, but not completely.

Place the bowl into your stand mixer and add the powdered sugar. Beat until well combined. Add in the salt, vanilla, and heavy cream and beat until it is a smooth, creamy consistency. Add more cream as needed to create your desired consistency.

Spread the frosting over the cooled banana cake. Slice and serve.