In a large heavy-bottomed pot, heat 1 tablespoon olive oil over medium-high heat. Brown the turkey in the pot about 6-7 minutes until no more pink remains; season with salt and pepper and drain on paper towels.
Add 2 tablespoons olive oil to the pot and saute the carrots, celery, and onion over medium-high heat until soft, about 7 minutes. Add the garlic and saute another minute until fragrant. Crush the tomatoes with your hand (or a wooden spoon) and add to the pot, along with the juices. Season with salt and pepper and bring to a boil. Reduce heat and let simmer, covered for 30-45 minutes. Stir a teaspoon or two of sugar, the Balsamic vinegar, and taste to see if it needs a bit more salt and pepper. Stir in the turkey and half of the basil. Remove from the heat.
Preheat the oven to 350 degrees. In a large skillet, heat 1 tablespoon of olive oil, and saute half of the spinach over medium-high for 3-4 minutes or until wilted. Season with salt and pepper; drain. Repeat the process with the remaining spinach and remaining olive oil. Let cool slightly.
In a medium bowl, stir together the ricotta, wilted spinach, and remaining basil. Season with salt and pepper to taste. Set aside. Combine the mozzarella and the Parmesan in a small bowl.
Cover the bottom of a 9×13″ ceramic or glass casserole dish with a thin layer of sauce. Make one layer of lasagna noodles on top of the sauce. Spread half of the ricotta mixture over the noodles and sprinkle with 1 cup of shredded cheese. Spoon half of the sauce over the cheese. Repeat the layers: noodles, ricotta, 1 cup cheese, and sauce. Top with the remaining cheese. (If you prefer to make 3 layers instead of 2, just layer the cheese and the sauce a little more thinly each time, and add an extra layer of noodles.)
Cover with aluminum foil and bake for 30 minutes. Remove foil and bake for another 15 minutes or until the cheese is melted and the sauce is bubbling. If you want, broil the top for 30 seconds to get it a little bit crispy!