Place the onions flat on the cutting board and cut into thick slices, allowing the root end to hold the slices in place.
Brush with some of the olive oil..just to coat and season with salt and pepper. Grill over a hot grill until streaked with grill marks, but still “crisp-tender.” Turn once. Remove to a cutting board, and cut into 1” pieces. Place in a bowl.
Cube large tomatoes or cut grape tomatoes in half. Add to onions. Add basil, remaining oil, vinegar and salt and pepper to taste.
Toss and serve.