Pass tomatoes through a food mill fitted with the large disk; set aside. Heat oil in a medium skillet over medium-low heat. Add garlic; cook until soft and light golden brown, about 2 minutes. Add reserved tomatoes, and simmer until thickened, about 35 minutes. Season with 2 teaspoons salt and 1 teaspoon pepper. Stir in torn basil leaves; set aside. In a large bowl, combine ricotta, butter, Parmesan, nutmeg, 1 teaspoon salt, and 1/2 teaspoon pepper. Stir until well combined. Set aside. Oil a rimmed baking sheet; set aside. Cover and bring a large pot of salted water to a boil. Add one sheet of pasta at a time to boiling water. Cook for 1 minute, and transfer to prepared baking sheet. Repeat until all pasta has been cooked. Preheat oven to 350 degrees F. Line a baking sheet with a silpat or parchment paper; set aside. Assemble the lasagna: Spread 1/2 cup tomato sauce in the bottom of a 9 by 13-inch baking pan. Place a layer of lasagna sheets in the pan, allowing sheets hang over the ends of pan. Spread 1/4 of the ricotta cheese mixture evenly over pasta and sprinkle with mozzarella. Top with a layer of pasta, cut to fit inside pan. Spread 1 cup tomato sauce over pasta. Repeat layering process, alternating pasta layers with cheese and sauce layers (there should be a total of 3 cheese layers and 2 sauce layers). Bring pasta hanging over ends of the pan up and over the top of final layer. Use more pasta, if necessary, to completely cover top. Spread with 1 1/2 to 2 cups tomato sauce. Dollop top of lasagna with remaining ricotta cheese mixture. Sprinkle with remaining mozzarella. Place on prepared baking sheet, and transfer to oven. Bake until the top is bubbling and light golden brown, 45 to 50 minutes; garnish with whole basil leaves.
In a medium skillet over medium heat, cook spinach, stirring occasionally, until wilted, 3 to 5 minutes. Remove from pan, and cool. Squeeze out excess liquid; there should be 1/4 packed cup of spinach. In the bowl of a food processor fitted with a steel blade, puree spinach until fine. Add 1 egg, and process few seconds more. Set aside. Mound flour in center of a work surface, and make a well in middle. Crack remaining 3 eggs into well. Beat eggs with a fork until smooth, then add spinach and egg mixture to well and stir until combined. Begin to work flour into egg mixture with fork. Use a bench scraper to work in the remaining flour, a bit at a time. Once all the flour has been incorporated, start working dough with hands to form a rounded mass for kneading. Be sure the work surface is clean of all loose bits of dough; lightly dust with flour. Knead dough until smooth and elastic, about 5 minutes. Cover dough with an inverted bowl or plastic wrap, and let rest 1 1/2 hours at room temperature, or refrigerate overnight. Divide dough into 8 pieces. Working with a few pieces at a time and keeping remaining pieces covered, flatten each portion of dough into a disk shape somewhat narrower than machine opening. Very lightly dust dough with flour. Feed through at pasta machine's widest setting, keeping in mind that machine settings differ; some have as many as 10, while others have only 6 . (If dough pulls or tears when passing through machine, sprinkle a little more flour over dough, just before it's fed, to keep it from sticking. When you're finished, remove excess flour with a dry brush.) As pasta sheet emerges, gently support it with palm, and guide it onto work surface. Fold sheet in 1/2. Pass through a second time on the same setting to smooth dough and increase its elasticity. Thin dough by passing it through ever-finer settings, 2 passes on each setting, until sheets are almost translucent but do not tear.