Heat sesame oil in large nonstick skillet over high heat. Add chicken; stir-fry 3 minutes. Remove.
Return skillet to heat; add vegetable oil; heat. Add mushrooms, scallions and carrots; stir-fry 5 minutes. Add stir-fry sauce and chicken to skillet; stir-fry until chicken is cooked through, 1 to 2 minutes. Top with cashews and serve with accompaniment.