Mash banana to a purée, making sure there are no chunks. Add lime juice and set aside.
Mix cake flour, baking soda, and salt. Set aside.
Cream butter and sugar together until crumbly and fluffy.
Beat in eggs one at a time to sugar and butter mixture. Add vanilla. Set aside.
Add vinegar to milk and let set for about a minute. Begin to add milk mixture and flour mixture to the creamed butter mixture, alternating between milk and flour.
Add bananas to cake mixture and blend well
Spray three 9" cake pans. Pour into pans and bake for 45 minutes or until cake tester comes clean.
To make frosting, cream butter and cream cheese together until well blended. Add vanilla.
Stir in powdered sugar. If you're wanting to frost the whole cake, add an extra cup of sugar to the mixture.
Once the cake is done baking, remove from oven and let set for 10 minutes then place in freezer for 35-40 minutes. I heard this is suppose to help it maintain a moist texture. After removing from the freezer, frost as desired