Heat the oven to 350°F. In a large skillet heat the olive oil. Add the zucchini, eggplant, mushrooms, spinach, and garlic, and sauté for about 5 minutes over medium-high heat, just to soften the vegetables slightly. Season with salt and pepper. Remove from the heat, and set aside.
Mix the goat cheese, basil, salt, and pepper until combined. Set aside.
Spread a thin layer of marinara sauce in the bottom of an 8-inch square baking dish. Top with a layer of noodles. Spread about 3 tablespoons of the goat cheese mixture over the noodles. Top with about ⅓ cup of the vegetable mixture.
Repeat layering with the remaining vegetables, noodles, goat cheese mixture, and marinara sauce. Cover tightly with foil, and bake for 40 to 50 minutes. Remove the foil, top with mozzarella, and broil for about 2 minutes, or until cheese is bubbly and golden brown.