Prepare barbecue (medium-high heat). Combine first 3 ingredients; add Asian sesame oil and toss to coat. Sprinkle with salt and pepper. Grill vegetables until tender, turning often, about 7 minutes for eggplant and mushrooms and 5 minutes for green onions. Transfer mushrooms and onions to cutting board; cool slightly. Slice mushrooms and onions. Meanwhile, cook noodles in large pot of boiling salted water until almost tender but still firm to bite, about 3 minutes. Add sugar snap peas to noodles; cook 1 minute longer. Drain well. Transfer noodle mixture to large serving bowl. Add grilled vegetables. Whisk last 4 ingredients in small bowl to blend. Toss noodle salad with dressing; season with salt and pepper. Test Kitchen Tip: The cooking time of Japanese-style chuka soba noodles varies by a minute or two depending on the brand of noodle. Test them after three minutes to avoid overcooking.