Heat a broiler or barbecue grill until it is hot. Position the rack about 4 inches from the heat. Brush the eggplant slices liberally with olive oil on both sides. Place them directly on the grill or under the broiler and cook them until they are tender and lightly browned, turning once. Watch them carefully so they don't burn. Remove them from the grill and set them aside on paper towels to cool slightly.
Roast the peppers until they are totally charred. Place them in bags to steam, and then remove the skins, discard the membranes and seeds, and cut them into 1-inch strips.
Discard the eggplant stems. Combine the eggplant with the peppers and onion rings in a large, flat serving bowl. Crumble the goat cheese on top.
In a small bowl, whisk together the balsamic vinegar, olive oil, thyme, salt and pepper. Drizzle the dressing over the eggplant mixture. Scatter the chopped parsley over the top. Add extra vinegar if needed.
This dish can be eaten immediately while still hot, or room temperature, and even the next day!
Makes 10 side-dish servings or 5 main-dish servings