Rinse zucchini, trim and discard ends, and shred.
In a 10- to 12-inch nonstick frying pan over high heat, stir oil, zucchini, onion, and oregano often until all the liquid has evaporated, about 6 minutes. Add salt and pepper to taste.
With a fork, beat eggs to blend with 1 tablespoon water.
Turn heat to low, pour egg mixture into pan and stir with a wide spatula, scraping pan bottom often, just until eggs are softly set, 2 to 3 minutes, or as firm as you like.
Spoon onto plates and sprinkle with cheese. Add tomato salsa, salt, and pepper to taste.