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.
I made this the other night and the sauce was a little thin. I also grated fresh ginger and think that the sauce would have made a better marinade turned sauce. Adding a little extra hoisin wouldn't hurt, either. I also omitted the veg oil and just did everything in the sesame oil. It's a good quick weeknight dinner.
I made this recipe as directed (with the exception of the garlic and ginger - I used fresh) and it was not that good. I think this was due to the preparation. If I try it again I will add the sesame oil to the vegetable oil when sauteeing the chicken and also throw the garlic and ginger in there instead of adding it to the broth mixture. I think this would give the whole dish more flavor.