To prepare chicken, combine first 3 ingredients in a medium bowl. Cover and marinate in refrigerator 20 minutes.
To prepare the sauce, combine cornstarch and water in a medium bowl. Stir in 1/3 cup broth and next 4 ingredients (1/3 cup broth through sesame oil).
To prepare stir-fry, heat vegetable oil in a large nonstick skillet over medium-high heat. Add chicken mixture; stir-fry 3 minutes or until chicken loses its pink color. Remove from pan. Add broccoli, bell pepper, and ginger; stir-fry 30 seconds. Stir in 3 tablespoons broth, and stir-fry 2 minutes. Stir sauce with a whisk until blended; add sauce and chicken mixture to pan. Bring to a boil; cook 2 minutes or until vegetables are crisp-tender and chicken is done. Serve over rice.
You can find jarred black bean sauce in the Asian-food sections of grocery stores or in Asian markets.