1. Preheat oven to 350°F. In large bowl, with mixer at high speed, beat egg whites until stiff peaks form. In small bowl, with same beaters and with mixer at high speed, beat egg yolks, water, and 1/2 teaspoon salt until very thick. With wire whisk or rubber spatula, gently fold egg-yolk mixture into beaten egg whites.
2. In 12-inch oven-safe skillet over medium-low heat, in 2 tablespoons hot salad oil, cook egg mixture until top is puffy and bottom is golden when gently lifted with a pancake turner, about 3 minutes.
3. Place skillet in oven. Bake until top of omelet is golden and springs back when lightly touched with finger, about 10 minutes.
4. Meanwhile, in 10-inch skillet over medium heat, in 2 tablespoons hot salad oil, cook zucchini, yellow straightneck squash, oregano leaves, and 1/2 teaspoon salt until vegetables are tender, stirring occasionally. Remove skillet from heat; keep warm.