Preheat oven to 350 degrees F. Thoroughly grease about 24 muffin cups.
In a large mixing bowl, beat together the egg and mayonnaise. Add the ripe bananas and beat until smooth. You can do this by hand or preferably, with a handheld electric mixer.
Beat in the sugar, brown sugar, vanilla, baking soda and salt. Stir in the flour.
Divide the batter between muffin cups (it should be about 2/3 of the way full) and bake for about 25 minutes or until muffins are set. Let cool slightly, then turn from pan. The muffins have flat tops, so I like to serve them upside down with a dab of cream cheese icing. To make as a loaf, bake in a 9x5 inch baking pan for 1 hour.
I don't mash my bananas separately, I just throw them in with the egg and mayonnaise and mash them with the electric mixer.