In a saucepan, mix together sauce ingredients over medium heat. Bring to a boil, and turn down to a simmer–cook until mixture reduces by about 1/4. Cool to about room temperature, and add chicken, scallions and mushrooms.
Allow to marinate at least three hours or so. Overnight would probably be even better.
Thread chicken and vegetables, alternating, on skewers of choice; if you use bamboo skewers, be certain to soak them in water to cover for at least three hours, preferably overnight. Reserve marinade.
Bring marinade back to a boil and allow to boil vigorously for about two minutes, in order to kill any bacteria that may have been introduced to the mixture by the chicken.
Grill skewers over high heat, basting with boiled marinade and turning frequently, until chicken has firmed up and is streaked with golden brown and black, and the mushrooms and scallions have blackened considerably.
Drizzle with sesame oil.
Serve immediately, on the skewers or off.