Oil grill grates (this oil is not listed in ingredients). Then preheat grill to medium high heat.
While heating, core and quarter your peppers, thickly slice red onion into rings and slice zucchini and squash lengthwise.
Working in batches, grill peppers, onions, zucchini and squash, flipping once, until just charred in parts and tender, 6 to 8 minutes total; transfer to a large platter as done. Set aside to let cool before cutting into bite-size pieces.
In a large bowl, combine grilled vegetables, chopped garlic, vinegar, oil, chopped basil, chickpeas, salt and pepper to taste. Garnish with 1/2 cup Parmesan and serve immediately or cover and chill until ready to serve.