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1
Heat 2 teaspoons of the oil in an extra-deep 12-inch skillet over medium heat.
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2
Peel and coarsely chop the onion, adding it to the skillet as you chop.
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3
(If you are using mushrooms, add them now.)
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4
Cook the onion, stirring frequently, until it is soft, 2 minutes.
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5
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).
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6
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.
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7
Set aside.
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8
When the vegetables are tender, transfer them to a small bowl and set it aside.
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9
Discard any juices that may have accumulated in the skillet.
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10
Add the remaining 2 teaspoons oil to the skillet and heat it over medium heat.
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11
Reduce the heat to medium-low.
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12
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.
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13
Turn the heat to low and sprinkle the Italian seasoning (if using) over one half of the omelet.
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14
Drain off any juices that may have accumulated around the cooked vegetables and sprinkle the vegetables evenly over half of the omelet.
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15
Sprinkle the cheese over the vegetables and season with salt and black pepper.
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16
Using a wide metal spatula, fold the plain half of the omelet over the filled half.
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17
Then cover the skillet and cook until the cheese melts, 1 to 2 minutes.
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18
Cut the omelet in half and serve.
