Rinse unripe tomatoes. Cut crosswise into 1/2-inch-thick rounds, discarding stem ends and bottoms. In a large bowl, gently mix tomato slices, olive oil, half the garlic, and the marjoram. Add salt and pepper to taste. Cover and chill at least 30 minutes or up to 1 day.
Meanwhile, rinse and core the ripe tomatoes; cut in half crosswise. Squeeze tomato halves slightly to remove seeds. Chop tomatoes and put in a bowl. Add the remaining garlic and the onion, red pepper, basil, lime juice, and chile. Add salt and pepper to taste.
Lift unripe tomatoes from marinade and lay on an oiled grill over a solid bed of medium coals or medium heat on a gas grill (you can hold your hand at grill level only 4 to 5 seconds); cover gas grill. Cook, turning once, until tomatoes are nicely browned, 6 to 10 minutes total. Arrange on a platter and spoon the ripe-tomato salsa over the top.
Note: Nutritional analysis is per serving.