In another bowl, whisk together the eggs, milk, vanilla extract, and the 2 tablespoons of oil. Combine this with the sugar-flour mix. Stir until all the flour is moistened.
Place bananas in bowl with batter, coat well. With fingers, drop bananas into at least 2 inches of heated (375 degree/190 degree C vegetable oil on deep frying thermometer) in a deep frying pan or a deep fryer.