Make the filling: in a small nonstick skillet over medium heat, saute pancetta for 2 minutes; add in green onions and bell pepper; stir/saute until pancetta is crisp and vegetables are tender, about 5 minutes.
Stir in 3 tablespoons beans and basil; transfer mixture using a slotted spoon to a plate and set aside; leave drippings in pan.
Make the omelet: in a bowl, whisk the eggs, milk, salt, pepper, and half the mozarella together.
Add olive oil to skillet with drippings; warm over medium heat.
When oil is hot, add egg mixture all at once; let set until edges start to cook, about 20 seconds.
With a spatula, gently lift edges of mixture and tip the skillet to allow the uncooked egg mixture to flow underneath.
Continue to do this until the top is almost dry, 3-4 minutes.
Spoon filling over one half of the omelet; sprinkle with the remaining mozzarella cheese.
Fold the other side over to cover; let stand a few seconds; turn out onto a warmed plate and sprinkle with Parmesan.
Cut in half and serve; heat remaining beans in a pan with a small amount of water and serve as a side dish.