Place an inverted baking sheet on a rack in the lower third of the oven and preheat to 500 degrees . Place a second inverted baking sheet on a work surface and cover with parchment.
In a nonstick skillet, heat 2 tablespoons olive oil over medium heat. Add the eggplant, season with salt and cook, stirring occasionally, until tender, about 10 minutes; transfer to a bowl. Add 1 teaspoon olive oil and the bell pepper to the skillet and cook until crisp-tender; add to the eggplant.
Meanwhile, using a food processor, puree three-quarters of the tomatoes, the garlic and thyme; season with salt and pepper. Transfer to the nonstick skillet and simmer over medium heat until reduced to about 1/2 cup, about 5 minutes. Halve the remaining tomatoes and place in a bowl. Add the zucchini and remaining 2 teaspoons olive oil; season with salt and pepper and toss to coat.
Stretch the pizza dough into 4 rectangles sized to fit the parchment, leaving a little space between each rectangle. Top with the cheese, tomato sauce and vegetables. Transfer the pizzas on the parchment to the preheated baking sheet in the oven. Bake until golden and crisp, 15 to 20 minutes.