Preheat your oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan.
In a bowl, whisk together the flour, baking soda, and salt. Set aside.
In another bowl, beat the butter and sugar until light and fluffy. Add the eggs one at a time, beating well after each addition.
Mix in the mashed bananas, sour cream, and vanilla extract until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Pour the batter into the prepared baking pan and spread evenly.
Bake for 30 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool completely in the pan on a wire rack.
For the frosting, brown the butter in a saucepan over medium heat, then let it cool slightly.
In a bowl, beat the browned butter and cream cheese until smooth. Gradually add the powdered sugar and vanilla extract, beating until the frosting is light and fluffy.
Spread the frosting over the cooled cake. Slice and serve.