Heat gas or charcoal grill. In Dutch oven or 4-quart saucepan, cook linguine to desired doneness as directed on package.
Meanwhile, in large bowl, mix tomatoes, basil, 4 tablespoons of the oil, the salt and pepper; set aside.
In small bowl, mix remaining 2 tablespoons oil and the garlic. In ungreased 15x10x1-inch pan, place eggplant, bell pepper, summer squash and onion. Brush with oil-garlic mixture.
Place eggplant, bell pepper, summer squash and onion pieces directly on grill over medium heat. Cover grill; cook 8 to 12 minutes, turning frequently, until vegetables are crisp-tender.
Coarsely chop vegetables; add to tomato mixture. Add chèvre; mix gently.
Drain linguine; place on serving platter. Top with vegetable mixture and Parmesan cheese.