COOK pasta as directed on package.
MEANWHILE, heat dressing in large skillet with tight-fitting lid on medium-high heat. Add onions and squash; cook 5 min., stirring frequently. Stir in mushrooms and Brussels sprouts; cook an additional 5 min., stirring frequently. Add chicken stock; stir until well blended. Bring to boil. Reduce heat to medium-low; cover. Simmer 5 min. or until vegetables are tender.
DRAIN pasta; place in large serving bowl. Add vegetable mixture and 1/4 cup of the cheese; toss lightly. Sprinkle with remaining cheese. Serve warm.