For sauce, in a small bowl stir together water, soy sauce, hoisin sauce, cornstarch, ginger, and sesame oil. Set aside.
Cut flowerets from broccoli stems and separate flowerets into small pieces. Cut broccoli stems crosswise into 1/4-inch slices. Cut pepper into short, thin strips.
In a wok or large skillet heat 1 tablespoon of the cooking oil over medium-high heat. Cook and stir broccoli stems in hot oil for 1 minute. Add broccoli flowerets and sweet pepper; cook and stir for 3 to 4 minutes or until crisp-tender. Remove from wok; set aside.
Add remaining oil to wok or skillet. Add chicken; cook and stir for 2 to 3 minutes or until no longer pink. Push chicken from center of wok. Stir sauce; pour into center of wok. Cook and stir until thickened and bubbly. Return cooked vegetables to wok. Stir together to coat. Cook and stir 1 minute more or until heated through. Serve over chow mein noodles or rice. If desired, garnish with toasted sesame seed and serve with additional hoisin sauce.