In a large bowl, combine chicken, oyster sauce and soy sauce. Set aside for 15 minutes.
Heat oil in a wok or large heavy skillet over medium heat. Saute garlic and onion until soft and translucent. Increase heat to high. Add chicken and marinade, then stir-fry until light golden brown, about 10 minutes. Stir in water, pepper and sugar. Add carrots and broccoli, and cook stirring until soft, about 10 minutes. Pour in the cornstarch mixture, and cook until sauce is thickened, about 5 minutes.