To make the basting sauce, combine all of the ingredients in a small saucepan and mix well. Set aside.
Preheat a gas grill to medium-high.
To make the patties, toast the sesame seeds on medium heat in a dry pan on the store.
Let cool and then combine with the chicken, hoisin sauce, cilantro, green onion, ginger juice, tamari, cornstarch, garlic powder, sesame oil, chile oil and pepper in a large bowl; handling the meat as little as possible to avoid compacting it, mix well.  Divide the mixture into 6 equal portions and form the portions into patties to fit the rolls.
When the grill is ready, brush the grill rack with the vegetable oil.  Place the patties on the rack, cover and cook, turning once and basting frequently with the sauce, just until the juices run clear when the patties are pierced in the center, about 5 minutes on each side.
Brush the mushrooms with the sesame oil and grill alongside the patties, turning frequently and brushing with the sauce until tender, about 10 minutes.
During the last few minutes of cooking, brush the cut side of the rolls with melted butter and place, cut side down, on the outer edges of the rack to toast lightly.
Place the saucepan on the stove and bring the remaining basting sauce to a boil.
To assemble the burgers, cut each mushroom into 4 wedges. Brush the basting sauce on the cut sides of the rolls. On each roll bottom, place a patty and 4 mushroom wedges. Add the roll tops and serve.