How to Make It
In a food processor, pulse the flour with a pinch of salt. With the machine on, add the eggs, egg yolks and olive oil and process until the dough resembles coarse sand. Transfer the dough to a work surface and knead until smooth. Flatten the dough into a disk, wrap it in plastic and refrigerate for at least 30 minutes or up to 6 hours.
In a medium saucepan, melt 2 tablespoons of the butter. Stir in the flour. Gradually whisk in the milk until the sauce is smooth, then bring to a boil over moderately high heat, whisking constantly. Reduce the heat to low and cook the sauce, whisking often, until no floury taste remains, about 10 minutes. Whisk in the cream and simmer for 3 minutes. Remove from the heat and stir in the proscuitto. Season the sauce with salt and pepper.
In a large skillet, heat the olive oil. Add the porcini, season with salt and pepper and cook undisturbed over moderately high heat for 1 minute. Stir well. Reduce the heat to moderate and cook until the mushrooms release their juices, about 5 minutes. Continue cooking until all the juices have evaporated and the mushrooms begin to brown, about 5 minutes longer. Add the garlic and cook over moderately high heat until fragrant, about 2 minutes. Add the rosemary and season with salt and pepper.
Preheat the oven to 425°. Bring a large pot of salted water to a boil. Set a very large bowl of cold water nearby. Cut the pasta dough into 4 pieces and keep 3 of them wrapped. Run 1 piece of the pasta dough through a pasta machine, beginning at the thickest setting and working your way through successively narrower settings until you reach the thinnest. Cut the pasta sheet in half and drape it over a drying rack or the back of a chair. Repeat with the remaining 3 pieces of pasta dough.
Cut the pasta sheets into 4-inch lengths. Cook half of the pasta in the boiling water for 1 minute. With 2 slotted spoons, transfer the cooked pasta to the bowl of water to cool for 2 minutes. Drain the pasta on paper towels. Repeat with the remaining pasta.
Butter a 10-by-14-by-1 1/2-inch glass or ceramic baking dish. Cover the bottom of the dish with a double layer of pasta. Spread 1 cup of the proscuitto sauce over the pasta and scatter one third of the mushrooms and 1/4 cup of the Parmesan on top. Cover with a layer of pasta. Repeat the layering 2 more times, ending with a layer of pasta. Spread the remaining 1/2 cup of prosciutto sauce over the top and sprinkle with the remaining 1/4 cup of Parmesan. Drizzle with the melted butter.
Bake the lasagne until bubbling and golden brown on top, about 25 minutes. Let the lasagna rest for 10 to 20 minutes before serving.