For dressing, in a food processor bowl or blender container combine rice wine vinegar, shallot, honey, soy sauce, garlic, sesame oil, and white pepper. With machine running, slowly add canola oil. Set aside.
Trim roots and green leaves from leeks. In a small saucepan cook leeks in boiling water about 8 minutes or just until tender; drain and halve lengthwise. Insert a wooden pick through one end of each leek half to hold it together while grilling. Remove and discard stems from mushrooms. Rinse mushroom caps; pat dry with paper towels. Place in a grill basket.
Preheat gas grill. Reduce heat to medium. Place leeks and mushrooms on the grill rack directly over heat. Cover and grill for 5 to 8 minutes or until vegetables are tender, turning once halfway through grilling. Remove from grill; cool slightly.
Coarsely slice the vegetables. Toss the vegetables with dressing. If desired, serve the vegetable mixture on lettuce leaves. Sprinkle with sesame seed. Makes 6 side-dish servings.
To use charcoal grill, prepare vegetables as above; grill over medium coals for 5 to 8 minutes.