Cut chicken into 1/4-inch-thick strips.
Coat a medium nonstick skillet or stir-fry pan with cooking spray; add oil. Place over medium-high heat until hot; add chicken, and stir-fry 2 minutes. Add broccoli and next 3 ingredients; stir-fry 2 minutes. Add bean sprouts, snow peas, and water chestnuts; stir-fry 1 minute.
Combine chicken broth and next 4 ingredients, stirring until smooth; add to chicken mixture. Cook, stirring constantly, until mixture is slightly thickened and thoroughly heated. Spoon chicken mixture over cooked rice.