Saute mushrooms and onion in 1 tablespoon oil in a large skillet until tender. Stir in the teriyaki sauce, brown sugar, garlic powder and ginger; heat through. Remove and keep warm.
Flatten chicken to 1/4-in. thickness. Cook chicken in remaining oil in the same skillet over medium heat for 2-3 minutes on each side or until no longer pink.
Top with mushroom mixture and cheese. Cover and let stand until cheese is melted. Freeze option: Before adding cheese, cool chicken and mushroom sauce. Place in freezer containers and top with mushroom sauce. To use, partially thaw in refrigerator overnight. Slowly heat chicken through in a covered skillet until a thermometer inserted in chicken reads 165°, stirring occasionally and adding a little broth or water if necessary. Top with cheese and proceed as directed.