Preheat a grill to medium. Brush the mushrooms and polenta slices with 3 tablespoons olive oil and season with salt and pepper. In a medium bowl, combine the eggplant and zucchini, drizzle with 3 tablespoons olive oil, season with salt and pepper and toss. Arrange the mushrooms, eggplant and zucchini on the cooking grate, cover and grill, turning once, until the mushrooms are tender and the eggplant and zucchini are golden, about 5 minutes; thinly slice the mushrooms. Add the polenta slices and grill until heated through, about 2 minutes.
Meanwhile, in a large pot of boiling, salted water, cook the pasta according to package directions until al dente; drain and toss with the chopped tomato and the remaining 2 tablespoons olive oil. Season with salt and pepper.
In a small bowl, combine the mozzarella and mayonnaise. Line a rimmed baking sheet with foil and grease the foil. Divide the eggplant into 4 stacks, then layer each stack with 3 zucchini slices, 3 tomato slices, some of the cheese mixture, 3 polenta rounds, more of the cheese mixture and some of the sliced mushrooms. Transfer the baking sheet to the cooking grate, cover and grill until the cheese is melted, about 5 minutes. Serve with the pasta.