In a bowl, whisk together 2/3 cup olive oil, soy sauce, vinegar, mustard, and garlic. Reserve 1/2 cup marinade for basting vegetables; pour remaining into a 1-gallon zip-lock plastic bag. Cut chicken lengthwise into 1-inch-wide strips and add to bag. Seal bag and turn to coat. Chill about 45 minutes.
Meanwhile, rinse zucchini, bell peppers, and mushroom caps. Slice zucchini lengthwise into 1/4-inch-thick strips. Stem and seed bell peppers; quarter lengthwise. Peel onion and slice crosswise into 1/4-inch-thick rounds. Brush vegetables with reserved marinade.
Cooking in batches if necessary, lay vegetables and chicken (discard marinade) on a barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid on gas grill. Cook, turning once, until vegetables are slightly charred and chicken is no longer pink in center of thickest part (cut to test), 6 to 10 minutes total. Let cool a few minutes. Chop vegetables and chicken into 1-inch pieces.
Meanwhile, in a 6- to 8-quart pan over high heat, bring 4 quarts water to a boil. Add pasta, stir, and cook, uncovered, until barely tender to bite, 9 to 13 minutes. Drain.
In a large bowl, mix pasta with remaining 1/3 cup olive oil, parmesan cheese, half the chopped vegetables and chicken, and salt and pepper to taste. Top with remaining vegetables and chicken. Garnish with sliced basil.