Heat 2 tablespoons of the olive oil in a large skillet over medium-high heat until hot. Add the mushrooms and cook, without moving them, until brown on 1 side, about 1 minute. Continue to saute for another 1 to 2 minutes. Season mushrooms with salt and pepper and pour onto cookie sheet to cool. Set used pan aside.
In a separate, clean skillet, heat 1 tablespoon oil over medium heat. Add minced garlic. Cook until golden brown, about 1 to 2 minutes. Add onion and cook until softened, about 5 minutes. In the pan where the mushrooms were, add 2 tablespoons olive oil over medium heat. Add yellow squash. Cook until browned, about 1 to 2 minutes and drain onto same cookie sheet with mushrooms.
Place peppers in hot pan where squash was cooking, sprinkle with 2 tablespoons of thyme. When heated thoroughly, add peppers to waiting mushrooms and yellow squash. Add zucchini to the same pan, add remaining thyme, cook until browned, about 2 minutes. Pull both pans (onions and zucchini) off the heat, and add to cookie sheet with other vegetables.
In the former zucchini/mushroom pan, cook leeks to soften, about 2 minutes. In the other, bring the tomatoes to a simmer. Season the leeks with salt and pepper, to taste, and add spinach. When spinach is wilted, add cooked leeks to cooling vegetables on cookie sheet. Add basil to the tomatoes, then add tomatoes to cookie sheet. Stir gently, and top with Parmesan and serve.