1. Preheat oven to 425 degree F and arrange racks in upper and lower thirds. Lightly coat a large shallow baking pan with cooking spray.
2. Wrap garlic in foil, sealing tightly, and place in one corner of prepared pan. Arrange tomato halves, cut sides up, in same pan; season with pepper. Roast in upper third of oven, without turning, until garlic is softened and tomatoes are golden brown, about 30 minutes.
3. Carefully unwrap garlic, peel and transfer to a food processor fitted with a blade. Add roasted tomatoes and 2-1/2 teaspoons rosemary; pulse until tomatoes are chopped. Transfer to a large saucepan and stir in crushed tomatoes with puree. Simmer, stirring occasionally, until slightly thickened and reduced to 6-3/4 cups, about 15 minutes, for sauce. Remove from heat and keep warm, covered.
4. Meanwhile, lightly coat another large shallow baking pan with cooking spray. Arrange mushrooms, in a single layer, in prepared pan and roast in lower third of oven, stirring occasionally, until golden brown and tender, about 15 minutes. Transfer mushrooms in pan to rack to cool.
5. Meanwhile, preheat broiler and arrange rack 5 to 6 inches from heat. Lightly coat a large shallow baking pan with cooking spray. Arrange half of eggplant slices, in a single layer, in prepared pan and season with pepper. Broil, turning once, until golden brown, about 6 minutes. Transfer to a plate and repeat with remaining eggplant slices.
6. Stir together ricotta, egg whites, and remaining 2 teaspoons rosemary in a bowl; season with pepper. Set aside.
7. Preheat oven to 375 degree F and arrange rack in center. Lightly coat a 13-by-9-by-2-inch baking dish with cooking spray. Evenly spread 1-1/2 cups sauce in bottom of dish and cover with 4 lasagna sheets, overlapping slightly if necessary. Evenly spread ricotta over lasagna and cover with half of mushrooms. Cover with half of eggplant slices, then spread evenly with 1-1/2 cups sauce. Continue layering with 4 lasagna sheets, 1 cup sauce, remaining mushrooms, and eggplant. Cover with remaining 4 lasagna sheets and top with remaining 2-3/4 cups sauce, spreading evenly. Arrange zucchini slices, in a single layer, over top.
8. Cover with nonstick foil and bake until lasagna sheets are tender and sauce is bubbling, about 40 minutes. Remove foil and sprinkle with Parmigiano-Reggiano; season with pepper. Bake, uncovered, until zucchini is golden and tender, about 15 minutes. Remove from oven; let stand 20 minutes before cutting and serving. Makes 10 servings.