Preheat oven to 350 degrees F. Cook pasta as directed on package. Meanwhile, spray large nonstick skillet with cooking spray. Add chicken, peppers and mushrooms; cook and stir 5 min. or until chicken is no longer pink. Remove from skillet; set aside.
Place Neufchatel, flour and broth in skillet. Cook and stir with wire whisk until boiling. Reduce heat to medium-low; simmer 5 min., stirring frequently.
Drain pasta; return to saucepan. Add Neufchatel cheese mixture, chicken mixture and 2 Tbsp. of the Parmesan cheese; mix well. Spoon into 2-qt. round casserole dish; cover with foil. Bake 25 min. Top with mozzarella and remaining 1 Tbsp. Parmesan cheese; continue baking, uncovered, 2 min. or until mozzarella cheese is melted.