Heat the flavored oil in a large nonstick skillet or wok. Add the garlic; saute for 30 seconds. Add the eggplant, onion, string beans, carrot and ginger; cook for 8 minutes or until tender. Add broth as needed to keep vegetables from sticking.
Meanwhile, combine the soy sauce, cornstarch and sugar in a small bowl. Add to vegetables; heat until the sauce thickens. Remove the skillet from the heat; keep warm.
Coat a 10-inch skillet, preferably nonstick, with nonstick vegetable- oil cooking spray. Beat together the whole eggs, egg white and sherry in a small bowl. Heat the skillet over medium heat. Add the egg mixture. Lower the heat; cook until eggs are almost set, about 2 minutes. Using a large pancake turner, carefully flip the omelet over. Cook 1 to 2 minutes longer or until lightly golden. Place on cutting board; thinly slice.
Serve the vegetables over rice and top with omelet shreds.