Preheat grill to medium-high. Brush tomatoes with oil; season with salt and pepper. Place on grill, cut-side up; cook until marked on bottom, about 5 minutes. Chop and transfer to a bowl. Stir in oil, vinegar, and basil; season with salt and pepper.
Brush onion, mushroom caps, and squashes with oil; season with salt and pepper. Grill until tender and marked in spots, about 8 minutes for onion and squashes, 7 minutes for mushrooms. Meanwhile, grill cheese until softened and marked, about 1 minute a side.
Divide romaine hearts, radicchio, vegetables, and cheese among plates. Drizzle with dressing, top with basil, and serve immediately.