Combine the chicken, oyster sauce, and soy sauce in a mixing bowl until the chicken is evenly coated with the sauce; set aside.
Heat the vegetable oil in a wok or large skillet over high heat. Stir in the garlic and onion; cook and stir until the onion is limp, about 1 minute. Add the chicken and marinade. Cook and stir until the chicken has browned and is no longer pink, about 10 minutes.
Pour in 1/2 cup of water; season with pepper and sugar. Add the water chestnuts, snow peas, and broccoli. Cover; boil until the vegetables are just tender, about 5 minutes. Dissolve the cornstarch in 1/4 cup of water. Stir into the boiling mixture; cook until thick and no longer cloudy.