In a 9- by 13-inch baking pan, combine onion and oil, then mix in bread. Bake in a 400 degree F oven until bread is golden, 10 to 12 minutes. Stir in tomatoes, vinegar, and minced marjoram and spread evenly in pan. Bake until crisp to touch, about 8 minutes; stir halfway through baking. Keep warm.
Meanwhile, in a bowl, whisk egg whites, eggs, water, and rosemary until well blended. Place a 7- to 8-inch nonstick frying pan over medium heat. Add half of egg mixture. As eggs cook, gently lift cooked portion to allow uncooked portion to flow underneath, about 2 minutes; then let stand on burner until top is set, about 1 minute more.
Spoon half of bread mixture onto half of omelet, then turn other half of omelet over bread to cover. Slide onto a plate. Repeat to make second omelet. Garnish with marjoram sprigs.