1. Heat 2 teaspoons of the oil in an extra-deep 12-inch skillet over medium heat. Peel and coarsely chop the onion, adding it to the skillet as you chop. (If you are using mushrooms, add them now.) Cook the onion, stirring frequently, until it is soft, 2 minutes.

2. When the onion is soft, add the garlic and the other vegetables to the skillet and cook until they are tender, 2 to 3 minutes (see Notes).

3. While the vegetables cook, break the eggs into a 1-quart or larger bowl, add 1 tablespoon water, and beat with a whisk or a fork until foamy. Set aside.

4. When the vegetables are tender, transfer them to a small bowl and set it aside. Discard any juices that may have accumulated in the skillet.

5. Add the remaining 2 teaspoons oil to the skillet and heat it over medium heat. Reduce the heat to medium-low. Pour the eggs into the skillet and cook without stirring until the edges are set and the middle is only slightly runny, 3 to 4 minutes. Turn the heat to low and sprinkle the Italian seasoning (if using) over one half of the omelet. Drain off any juices that may have accumulated around the cooked vegetables and sprinkle the vegetables evenly over half of the omelet. Sprinkle the cheese over the vegetables and season with salt and black pepper.

6. Using a wide metal spatula, fold the plain half of the omelet over the filled half. Then cover the skillet and cook until the cheese melts, 1 to 2 minutes. Cut the omelet in half and serve.