For carrot layer: Cook frozen carrots in large saucepan of boiling water until soft, about 15 minutes. Drain. Melt butter in heavy large skillet over medium-high heat. Add onion and saute 3 minutes. Puree carrots with onion and ricotta in processor. Add salt and pepper. (Can be prepared 1 day ahead. Chill.) For spinach layer: Cook spinach according to package instructions. Drain and cool. Squeeze out excess water. Transfer to processor. Heat oil in heavy small skillet over medium-high heat. Add shallots and saute 2 minutes. Add to processor. Add ricotta and egg to processor and puree mixture. Season with salt and pepper. (Can be prepared 1 day ahead. Cover and refrigerate.) For eggplant layer: Salt eggplant and place between paper towels. Let stand 30 minutes. Heat 2 tablespoons oil and 1 minced garlic clove in heavy large skillet over medium heat. Add eggplant in single layer and cook until tender and golden, turning occasionally, about 10 minutes. Transfer to paper towels to drain. Repeat with remaining oil, garlic and eggplant in 2 more batches. Season with pepper. Cook noodles in large pot of boiling salted water until just tender but still firm to bite, stirring occasionally. Drain. Rinse under cold water to cool; drain. Butter 9x13x2-inch baking dish. Spread 1/2 cup tomato sauce on bottom of dish. Arrange 1/4 of noodles over. Season with salt and pepper. Spread carrot puree over. Spoon 1/2 cup tomato sauce over. Sprinkle with 1 tablespoon basil, 1 cup mozzarella and 3/4 cup Parmesan cheese. Top with 1/4 of noodles. Season with salt and pepper. Spread spinach mixture over. Spread 1/2 cup tomato sauce over. Sprinkle with 1 tablespoon basil, 1 cup mozzarella and 3/4 cup Parmesan cheese. Top with 1/4 of noodles. Season with salt and pepper. Arrange eggplant in overlapping slices atop noodles. Spread with 1/2 cup tomato sauce. Sprinkle with 1 tablespoon basil, 1 cup mozzarella and 3/4 cup Parmesan cheese. Top with remaining 1/4 of noodles. Season with salt and pepper. Spread 1/2 cup tomato sauce over. Sprinkle with remaining 1 tablespoon basil, 1 cup mozzarella and 3/4 cup Parmesan cheese. (Can be prepared 1 day ahead. Cover and refrigerate. Let stand 2 hours at room temperature before continuing.) Preheat oven to 350°F. Bake lasagna until heated through, about 1 hour 15 minutes. Cool 10 minutes before serving.