Combine oyster sauce, sesame oil, Vermouth, soy sauce, corn starch, garlic and ginger in a small bowl.
Pat dry chicken breasts with paper towel. Thinly slice into strips. Add to oyster sauce mixture. Stir to coat; marinate for at least 30 minutes in refrigerator.
Add 1 Tablespoon oil to wok; heat oil in wok over high heat. Add onion and broccoli; toss and stir in the hot oil until broccoli turns bright green and is almost tender, and onions are lightly browned, 5 to 7 minutes. Remove broccoli and onions from wok; set aside.
Add remaining oil to wok. Add chicken and marinade; stir and toss the chicken with the marinade until the sauce forms a glaze on the chicken and the chicken is no longer pink, 2 to 3 minutes. Return cooked broccoli and onions to wok and stir until the chicken and vegetables are heated through, about 2 minutes. Serve immediately with hot white rice. Yield: 2 servings.