1. In 8- to 12-quart saucepot over high heat, in 2 inches boiling water, heat onions, peppers, zucchini, yellow squash, eggplant, and 2 tea spoons salt to boiling. Reduce heat to low; cover and simmer 5 minutes or until peppers, zucchini, and yellow squash are tender-crisp, stirring occasionally. With tongs, remove peppers, zucchini, and squash to large roasting pan. Cover saucepot; cook 20 to 25 minutes longer, until onions and eggplant are fork-tender; remove to same roasting pan.
2. Cut each onion, pepper, zucchini, and yellow squash lengthwise in half; discard seeds from pepper. Cut eggplant crosswise in half, then cut each half lengthwise into eighths. Return parboiled vegetables with mushrooms to roasting pan; add oil, vinegar, sugar, rosemary, thyme, and 2 teaspoons salt; toss to coat well with marinade. Cover and refrigerate at least 4 hours, tossing vegetables occasionally.
3. About 45 minutes before serving, prepare outdoor grill for barbecuing.
4. Place vegetables on grill over low heat; cook 10 minutes, turning vegetables occasionally, until heated through, basting occasionally with marinade in roasting pan.