Peel the onion and chop finely. Roughly dice the vegetables or cut them into bite-sized pieces. Put some oil in a small pan and briefly fry the onion and vegetables with a little salt and a little sugar. Remove the onion and vegetable mixture from the pan and set aside.
Whisk both eggs with two tablespoons of water, a little salt, pepper and thyme as well as the cheese. Put the egg mixture with a little oil in the small pan and place the vegetables on the egg.
Cook over low heat until the egg is no longer runny on the top. I always put a lid on the pan and turn the stove down very low so the omelette doesn`t burn from below.
Tip:
I like to remove the vegetables with a little water after frying them. This makes a great brew that I can add to the omelette at the end of the cooking time so that it doesn`t burn from below. If you like the omelette even more loosely, you can add some cream or milk or even more water instead of the water. It`s a matter of taste.
My favorite vegetables are broccoli, peppers and leeks. With the leek, I can save on the onion. But everything else is also suitable. What is just there and has to go if necessary.