In a large resealable plastic bag, combine the oil, salt and pepper. Add the asparagus, squash, red pepper, onion and mushrooms; seal bag and turn to coat.
Transfer the vegetables to a grill wok or basket. Grill, uncovered, over medium heat for 8-12 minutes or until tender, stirring frequently.
Place vegetables in a serving bowl. Add vinegar and basil; toss to coat. Sprinkle with Parmesan cheese. Serve warm.
Editor's Note
If you do not have a grill wok or basket, use a disposable foil pan. Poke holes in the bottom of the pan with a meat fork to allow liquid to drain