1. Heat 1 tablespoon oil in a medium saucepan. Add onions and cook, stirring occasionally until softened, about 5 minutes. Stir in rice, then add broth and bring to a boil. Reduce heat to low, cover and cook until rice is tender, about 15 minutes.
2. In a large bowl, whisk remaining olive oil, parsley, vinegar, thyme, garlic, salt, and pepper. Add chicken, mushrooms, and tomatoes and toss until well coated. Thread alternating pieces of chicken, mushrooms, and tomatoes on skewers.
3. Heat broiler. Lightly oil a rack and place on a broiler pan. Arrange skewers on rack and broil 6 inches from heat source, 5 to 7 minutes. Turn skewers and continue to broil 5 minutes more, until chicken is cooked through. Serve skewers over rice.