Place cornstarch in a large bowl; gradually whisk in water until smooth. Whisk eggs into cornstarch mixture; toss steak strips in mixture to coat.
Pour canola oil into a wok, 1-inch deep; heat oil over high heat until hot but not smoking. Place 1/4 of the beef strips into hot oil; separate strips with a fork. Cook, stirring frequently, until coating is crisp and golden, about 3 minutes. Remove beef to drain on paper towels; repeat with remaining beef.
Drain off all but 1 tablespoon oil; cook and stir carrot, bell peppers, green onions, ginger, and garlic over high heat until lightly browned but still crisp, about 3 minutes.
Whisk sugar, rice vinegar, soy sauce, sesame oil, and red pepper flakes together in a small bowl. Pour sauce mixture over vegetables in wok; bring mixture to a boil. Stir beef back into vegetable mixture; cook and stir just until heated through, about 3 minutes.
