For salsa, in a medium bowl stir together tomatoes, cucumber, avocado, onion, garlic, parsley, vinegar and 1 teaspoon of the oil. Set aside (see Tip).
In a medium bowl, whisk together eggs, egg product, the water, oregano, salt, black pepper and crushed red pepper. For each omelet, in an 8-inch nonstick skillet heat ½ teaspoon of the remaining oil over medium heat. Add ½ cup of the egg mixture to skillet. Stir eggs with a spatula until mixture resembles cooked egg pieces surrounded by liquid. Stop stirring, but continue cooking until egg is set. Spoon ⅓ cup of the salsa over one side of the cooked egg mixture. Remove from skillet; fold omelet over the filling. Repeat to make four omelets total.
Serve each omelet with one-fourth of the remaining salsa. Sprinkle each omelet with 1 tablespoon of the feta cheese.
To make ahead: If desired, make the salsa ahead and cover and chill it for up to 8 hours.