In a skillet (preferably cast iron, of a certain size - see note #1), add half the quantity of oil you are planning to use, spread it over the skillet, and lightly heat it up for a minute or so.
Then pour the egg mixture in the skillet, and spread out the veggies evenly if you want to.
And then if you choose to add it, this is the time to add the shredded mozzarella cheese on top, before the mix starts getting cooked.
Then cover and cook for about 1-2 minutes on medium heat, till the cheese has melted, and the bottom has firmed up a little.
And then flip the egg omelette to the other side, and now add the rest of the oil from the sides.
Cook for another minute or so, till the egg is fully cooked, and then turn off the heat, and take the omelette off the pan.
Serve hot with regular sliced bread, Indian pav, or even roti.