Make the vegetables: Combine vinegar, oil, oregano, red-pepper flakes, and garlic. Pour over onion, peppers, and mushrooms on a rimmed baking sheet. Marinate at room temperature, turning occasionally, for 30 minutes.
Preheat grill to medium-high. Remove vegetables from marinade, and reserve marinade. Season vegetables with the salt and pepper; grill, turning and brushing occasionally with marinade, until tender, 5 to 6 minutes for onion, 10 to 12 minutes for peppers and mushrooms. Fill mushrooms with cheese; grill until cheese melts, about 30 seconds. Grill bread until toasted, about 30 seconds per side.
Make the salad: Whisk together oil, the salt and pepper, and 3 tablespoons reserved marinade; toss with greens. Serve with vegetables and bread.