Grill the vegetables for these toasts until browned and tender, and then toss with balsamic vinegar so the flavors and textures mix and meld for this open-face sandwich with mozzarella, charred peppers, and eggplants. Leftover vegetables would be delicious with pasta.
Prepare the grill and the vegetables: Heat a gas grill to medium high or build a medium charcoal fire. Set the eggplant and peppers on a large rimmed baking sheet, sprinkle generously with salt and pepper – about 1 tsp. salt and ½ tsp. pepper – and then drizzle with 2 Tbs. olive oil and toss well. Brush the bread with 2 Tbs. oil and sprinkle lightly with salt, about ½ tsp.
Grill the vegetables and bread: Set the vegetables on the grill and cook, undisturbed, until they brown and easily release when you lift their edges, 2 to 3 minutes. Flip and continue cooking until browned and tender, about 5 minutes total for the eggplant and 8 to 10 minutes for the peppers. Transfer to a large cutting board. As you pull some of the vegetables off the grill, add the bread slices. Cook until browned with good grill marks on both sides, about 4 minutes total.
Assemble and serve: Heat the broiler to high. Gently rub both sides of the toasts with the garlic and set on a large baking sheet. Coarsely dice the vegetables and toss with the remaining 2 Tbs. oil, the vinegar and the oregano. Top the bread with the vegetables and sprinkle lightly with salt. Add a slice of cheese and broil until the cheese melts. Drizzle with olive oil.