Mix chicken, egg, 2 teaspoons cornstarch, and sesame oil in a bowl.
Bring a large pot of water to a boil over high heat. Add noodles and cook until soft, about 4 minutes. Drain and rinse under cold water to stop the cooking.
Heat 1 tablespoon of vegetable oil in a wok over high heat. Add vegetables and stir-fry until soft, about 3 minutes. Transfer vegetables to a bowl using a slotted spoon.
Add the remaining vegetable oil to the wok. Add chicken and stir-fry until no longer pink, about 5 minutes. Transfer chicken to a bowl using a slotted spoon.
Mix the remaining cornstarch and water in a bowl until smooth; add to the wok. Add chicken broth, soy sauce, and oyster sauce, and cook over medium heat 3 minutes or until sauce starts to thicken.
Add vegetables and chicken back to the wok and stir until sauce is evenly distributed. Turn off heat and let stand 3 minutes. Serve over noodles in a bento box.