Dressing: In medium-size bowl, whisk together vinegar, mustard, salt and pepper. Drizzle in the olive oil, whisking until oil is thoroughly incorporated. Stir in garlic, thyme and oregano. Set aside half of the dressing for the pasta.
Vegetables and Pasta: Prepare outdoor grill with hot coals or heat gas grill. Lightly brush grill grid with vegetable oil. Generously brush peppers, zucchini, squash, onion and tomatoes with some of the remaining dressing.
Working in batches, grill the vegetables about 3 minutes per side or until fork-tender, about 18 minutes total. Brush the vegetables with additional dressing halfway through the grilling. Remove the vegetables from the grill and keep warm.
In large bowl or on serving platter, toss together cooked pasta with reserved half of dressing and any remaining dressing used for grilling vegetables. Toss in basil and mozzarella cheese. Arrange grilled vegetables over top. Serve warm or at room temperature.