Start by peeling the bananas, cut them into round slices about one-finger thick and set aside.
In a medium bowl, add the eggs, season them with a pinch of salt, a pinch of black pepper and mix everything thoroughly until combined.
Now, in a large skillet over medium to low heat, add the unsalted butter and stir until it melts.
Add a drizzle of vegetable oil and stir again to combine.
Next, add the banana slices we cut earlier and pour in the egg mixture.
Cover the skillet and let everything fry for 5 minutes.
After this time has passed, flip the omelet using a plate.
Place it back in the skillet and add the grated mozzarella cheese.
Cover the skillet again, let it fry for another 5 minutes or until the cheese is fully melted.
Serve.