Place onions in just enough water to cover. Bring to a boil; simmer five minutes. Drain; allow to cool. Squeeze onions to remove skins. Thread tomatoes, mushrooms, onions, and red peppers onto separate skewers. In a small bowl, whisk together olive oil, lemon juice, basil, oregano, garlic, salt, and pepper until well blended. Brush half the dressing on the skewered vegetables. (If you prefer, you can do this step ahead of time. Simply cover and refrigerate up to 12 hours.)
Grill vegetables over high heat until they are lightly charred but still tender crisp. Meanwhile, toss chicken and salad greens with remaining dressing. Divide salad among individual dinner plates. Remove vegetables from skewers. For kids 4 and under, slice mushrooms, onions, and tomatoes so they won't be a choking hazard. Divide to top salads.