Heat a large, deep nonstick skillet over high heat. Add the meat, season with 1/2 teaspoon of the salt, and cook, using a wooden spoon to break the meat into small pieces as it browns, 4 to 5 minutes.
Put the skillet over medium heat. Add the butter carrots and onion and cook, stirring, until soft, 3 to 4 minutes. Add the garlic and cook 1 minute. Return the meat to the pan, add the tomatoes, basil, 1/4 teaspoon of the salt, and the black pepper to taste. Reduce the heat to low, cover, and simmer, stirring occasionally, 25 minutes. Remove the lid and simmer uncovered 10 minutes, until thickened.
Meanwhile, slice the zucchini lengthwise with a mandolin into 1/8-inch-thick slices (you should have at least 30 to 35 long zucchini ribbons). Lightly salt the zucchini with the remaining 1/2 teaspoon salt and set aside for 15 minutes. Blot the zucchini with paper towels.
Preheat the oven to 375°F.
In a medium bowl, combine the ricotta, Parmesan, spinach and egg. Spread 1/2 cup of the meat sauce in the bottom of a 9 × 13 × 2 1/2- inch baking dish. Make a layer of the zucchini over the sauce to cover the bottom of the dish. Spread 1/2 cup of the ricotta mixture over the zucchini and sprinkle with 1 cup of the mozzarella. Make another layer of zucchini, top with 1 1/2 cups meat sauce, 1/2 cup ricotta mixture, 1 cup mozzarella. Repeat the layers with the remaining ingredients for a total of 3 layers. Finish the lasagna by topping with the remaining zucchini and meat sauce and a bunch of cheese on top! Bake for 1 hour. Let stand for 5 to 10 minutes before cutting into 8 pieces. I personally like to let it sit for a couple hours then re-heat. It allows some the the juices to absorb.