Heat vegetable oil in a large nonstick skillet coated with cooking spray over medium-high heat. Combine broth and the next 7 ingredients (broth through sesame oil). Add chicken to pan, and sauté 2 minutes. Add broccoli, and sauté 5 minutes. Add chestnuts, and sauté 2 minutes. Add broth mixture, and cook 1 1/2 minutes or until sauce is slightly thick, stirring constantly.