Mix the butter and sugars together until very well blended. Mix in the egg and beat thoroughly by hand. Add in the oil, bananas and vanilla and mix well.
In a separate bowl, mix all remaining dry ingredients.
Carefully mix dry ingredients into wet ingredients until just barely blended. You will still see some small flour patches in the batter. If you over-mix it, the muffins will become rubbery.
Line 12 muffin tins with paper cupcake liners and spray with nonstick cooking spray. Pour batter into the cups, filling almost to the top.
Bake for 20-25 minutes at 350-degrees, or until a knife just barely comes out clean when inserted into the center of a muffin.
While muffins are baking, mix all frosting ingredients well - it should be the consistency of buttercream frosting.
Once muffins are done, let cool for 10 minutes before removing fro the tin to cool the rest of the way.
Once cool, top with a thick layer of the frosting. Store in the refrigerator. Enjoy!