Preheat oven to 350°F. Grease and line a 9x13 pan with parchment paper. In a large bowl, whisk melted butter, oil, granulated sugar, and brown sugar.
Add eggs and vanilla; whisk until smooth. Mix in mashed bananas, buttermilk, and sour cream.
In a separate bowl, whisk flour, baking powder, baking soda, cinnamon, and salt.
Mix the dry ingredients into wet just until combined. Don’t over mix.
Pour the banana cake batter into the prepared pan and bake for 30-35 minutes, or until a toothpick comes out clean.
Cool completely before frosting.
Brown the butter in a light-colored saucepan over medium heat, stirring often, until it foams, smells nutty, and the milk solids turn golden brown, then immediately remove it from the heat to prevent burning. Note: watch the butter closely as it can burn quickly.
Immediately transfer the brown butter into a separate bowl. Let it cool slightly, then refrigerate just until it begins to slightly solidify.
In a large bowl, add the cooled brown butter, softened cream cheese, vanilla, and salt, and beat with an electric hand mixer until smooth.
Add vanilla, and salt; mix well.
Beat in powdered sugar until smooth and fluffy. Frost the banana cake, slice and enjoy!