How to Make It
Process flour, eggs, egg yolks, 1 tablespoon water, and oil in a food processor until mixture is evenly moistened, resembles couscous, and sticks together when pinched, adding remaining 1 tablespoon of water if needed to moisten dough, about 20 seconds. Turn dough out onto a clean work surface. Shape into a ball, and knead until smooth, about 3 minutes. Wrap dough tightly in plastic wrap. Let stand at room temperature at least 1 hours and 30 minutes or up to 4 hours.
Set rollers of a pasta machine on widest setting. Unwrap dough, and cut in half. Rewrap one dough half. Use a rolling pin or your hands, flatten unwrapped dough portion to 1/3-inch thickness. Roll dough through pasta machine, flouring dough as needed to prevent sticking. Fold dough in half crosswise; reroll through pasta machine. Continue folding dough in half crosswise and rerolling until it is as wide as the pasta rollers (5 to 6 inches), 1 or 2 times. Trim dough as needed to create straight sides. Once width is reached, run dough sheet through widest setting once more to ensure even thickness of dough. Reduce width of rollers one setting, and roll dough through rollers. Measure dough length, and trim to 32 inches, reserving scraps. Continue rerolling dough (do not refold) through pasta machine, reducing width of rollers one setting at a time and trimming to 32 inches between settings, until dough is the thickness of 3 pieces of paper and dough sheet weighs approximately 6 ounces. Trim pasta sheet into a 32- x 5-inch rectangle. Cover pasta sheet and scraps with plastic wrap or a towel. Repeat process with remaining dough half to form 2 (32- x 5-inch) sheets of pasta and about 6 ounces of dough scraps.
Bring a large pot of water to boil, and salt it. Add 1 pasta sheet to boiling water; cook 1 minute. Using a spider and tongs, gently transfer pasta to a baking sheet generously coated with oil. Coat pasta generously all over with oil to prevent sticking; fold in half crosswise. Repeat process using remaining pasta sheet. Set aside.
Process Parmigiano-Reggiano chunks in a food processor until finely ground into small crumbles, about 30 seconds. Transfer half of the cheese (about 3/4 cup) to a bowl, and reserve for finishing lasagna. Add ricotta, prosciutto, cream, and rosemary to food processor; pulse until well combined and prosciutto is very finely chopped, about 10 pulses. Season with salt to taste. Transfer filling to a large piping bag with a 3/4-inch hole snipped in the tip.
Coat a 9- x 5-inch loaf pan with oil. Unfold pasta sheets, and arrange them parallel, overlapping them to create a 9-inch-wide layer of pasta. Line the bottom and two long sides of the loaf pan with the pasta, allowing a 5-inch overhang on one side and the remaining long ends on the other.
Pipe a 1 1/4-inch-thick line of filling along the long pan side with the long ends of the pasta. Pipe a second line of filling parallel to the first, leaving 1/2 inch space on each side of the second line. Carefully fold the long overhanging pasta sheets up and over filling; press the pasta down to hug sides of filling. (This creates 2 long pockets of filling.) Pipe 2 lines of filling in the “trenches” of the pasta. Fold the longer pasta sheets back up and over the filling; gently press down to smooth.
Repeat Step 7 once for a total of 4 layers of filling. Fold the 5-inch pasta overhang up and over to seal the lasagna. Trim pasta sheets along upper lip of loaf pan. Wrap loaf pan in plastic wrap, pressing plastic wrap directly onto lasagna surface. Chill until set, at least 6 hours or up to 24 hours.
Bring a medium saucepan of water to a boil, and salt it. Add spinach; cook, stirring occasionally, until very tender, about 2 minutes. Using a spider, transfer spinach to a large bowl of ice water. Let cool 2 minutes. Drain and squeeze excess moisture from spinach. Transfer spinach to a blender; add 1/2 cup water. Process until smooth, about 30 seconds, adding up to an additional 1/3 cup water, 1 tablespoon at a time, as needed to thin sauce to consistency of heavy cream. Transfer mixture to a small saucepan. Stir in oil and salt. Keep warm over low, stirring often, until ready to use.
Preheat oven to 500°F. Run a small offset spatula around edges of lasagna to loosen from pan. Unmold lasagna, and invert onto a cutting board; cut crosswise into 6 (1 1/2-inch-wide) slices. Arrange lasagna slices, cut sides up, on a baking sheet lined with parchment paper, leaving at least 1 inch between slices. Brush tops and sides of slices evenly with oil. Sprinkle slices with reserved Parmigiano-Reggiano crumbles (about 2 tablespoons per slice). Bake in preheated oven until slices are heated through and edges are puffy and golden brown, 6 to 10 minutes.
Spoon 2 tablespoons spinach sauce onto each of 6 plates; tilt each plate in a circular motion to spread sauce in a thin, even layer. (Reserve remaining sauce for another use.) Place 1 lasagna slice on each plate. Serve immediately.
Prosciutto may be substituted with 1 cup of mushroom duxelles. To make duxelles, finely chop 1 lb. fresh mushrooms and saute in ¼ cup butter, stirring often, until mushrooms are cooked and all their liquid has evaporated, about 10 minutes. Season generously with salt and pepper, and let cool before adding to filling.