1. Preheat oven to 375 degrees F. Cook lasagna noodles according to package directions. Drain noodles; rinse with cold water. Drain again; set aside.
2. Meanwhile, in a large skillet, heat 1 tablespoon of the olive oil over medium-high heat. Add carrots, zucchini, and half of the garlic. Cook and stir about 5 minutes or until vegetables are crisp-tender. Transfer vegetable mixture to a bowl. Add the remaining oil to the same skillet and heat over medium-high heat. Add mushrooms and remaining garlic. Cook and stir about 5 minutes or until tender. Gradually add spinach. Cook and stir for 1 to 2 minutes or until spinach is wilted. Using a slotted spoon, transfer spinach-mushroom mixture from skillet to a bowl. Stir basil into spinach-mushroom mixture; set aside.
3. In a small bowl, stir together egg, ricotta cheese, Parmesan cheese, salt, and pepper. Set aside.
4. To assemble pie, in the bottom of a 9x3-inch springform pan, spread 1/2 cup of the pasta sauce. Arrange three to four of the cooked noodles over the sauce, trimming and overlapping as necessary to cover sauce with one layer. Top with half of the spinach-mushroom mixture. Spoon half of the ricotta cheese mixture over spinach mixture. Top with another layer of noodles. Spread with half of the remaining pasta sauce. Top with all of the vegetable mixture. Sprinkle with half the Fontina cheese. Top with another layer of noodles. Layer with remaining spinach-mushroom mixture and remaining ricotta cheese mixture. Top with another layer of noodles (may have extra noodles) and remaining sauce. Gently press down pie with the back of a spatula.
5. Place springform pan on a foil-lined baking sheet. Bake for 45 minutes. Sprinkle with remaining Fontina cheese; bake about 15 minutes more or until heated through. Cover and let stand on a wire rack for 15 minutes before serving. Carefully remove side of pan. To serve, cut into wedges. If desired, garnish with rosemary sprigs.