Mix 2 tablespoons each sliced scallion and chopped chestnuts with the duck sauce in small bowl.
Heat oil in medium-size nonstick skillet over medium-high heat. Add mushrooms, garlic and remaining scallion. Stir-fry until mushrooms are tender and scallion is soft, about 3 to 4 minutes. Let cool slightly.
Gently combine ground chicken, mushroom mixture, remaining chestnuts, chopped cilantro and soy sauce in a bowl. Shape into 4 patties about 3/4 inch thick. Season tops with half the salt and pepper.
Add patties to same skillet, seasoned side down. Sprinkle with remaining salt and pepper and cook over medium heat for 9 minutes. Gently turn patties. Cook 8 to 9 minutes longer, or until they register 165 degrees F on an instant-read thermometer.
Fill each bun with a burger. Top with sauce and cilantro sprig. Pass any extra sauce.