If you want to make only two grilled summery ratatouille pizza, use the remaining ratatouille in pasta, folded into eggs, or on a grilled cheese sandwich.
Prepare the grill and the vegetables: Heat a gas grill to medium-high or build a medium charcoal fire. Set the eggplant, peppers, zucchini, red onion, and summer squash on a large rimmed baking sheet (try to keep the onions together in disks so they grill easily), sprinkle generously with salt and pepper (about 1½ tsp. salt and 3/4 tsp. pepper) and then drizzle with ¼ cup olive oil. Toss well.
Grill the vegetables: 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 and flipping until the vegetables are browned and tender, about 5 minutes total for the eggplant, zucchini, and summer squash and 8 to 10 minutes for the peppers and onion. Transfer the grilled vegetables to a large cutting board.
Assemble and serve: Once the vegetables are cool enough to handle, coarsely chop them and transfer to a large serving platter. In a small cup, whisk together the remaining ¼ cup olive oil with the balsamic vinegar, thyme, and garlic. Drizzle over the vegetables and gently toss. Season with salt and pepper to taste, then sprinkle with the parsley.
Lightly brush the pizza shells with oil on both sides. Grill, top side down, until light grill marks appear, about 2 minutes. Flip the shell, then evenly cover each pizza with about 2½ cups of the vegetable filling. Top each pita evenly with 3½ oz. goat cheese. Continue to grill until just warmed through, about 2 minutes more. Remove from the grill and serve.