Place the tomatoes in a bowl and toss with 2 tablespoons of the oil and season with salt and pepper. Place tomatoes on the grill and grill until charred on all sides, and just cooked through, about 10 minutes. Remove from the grill and coarsely chop.
Put the chopped tomatoes in a bowl, add the remaining 3 tablespoons of oil, onion, vinegar and basil and gently mix until combined. Let sit at room temperature for 30 minutes before serving. Can be made 4 hours in advance and refrigerated. Bring to room temperature before serving.
Brush eggplant slices on both sides with the oil and season with salt and pepper. Place the slices on the grill and grill until golden brown and slighty charred, 4 to 5 minutes. Turn over and continue grilling until just cooked through, 3 to 4 minutes.
Stir together the ricotta and parsley in a medium bowl and season with salt and pepper.
Place the grilled eggplant on a large platter and top each slice with a heaping tablespoon of the ricotta and a heaping tablespoon of the tomato relish.