Make the ragu: Soak the porcini mushrooms in 2 cups hot water until soft, about 30 minutes. Drain the porcini, reserving the liquid.
Finely chop the porcini and white mushrooms in a food processor; transfer to a bowl. Add the carrots, celery, garlic, parsley and prosciutto to the processor; pulse until finely chopped.
Heat the olive oil in a large pot over high heat. Cook the ground beef and pork in batches, breaking them up with a wooden spoon, until browned, 6 to 8 minutes. Transfer the meat to a bowl. Add the wine to the pot, scraping up the browned bits. Add the liquid to the bowl with the meat.
Melt the butter in the pot; add the onion, chopped carrot mixture, the rosemary, bay leaves and 2 teaspoons salt. Cook, stirring occasionally, until browned, 12 to 15 minutes. Add the tomato paste and mushrooms and cook, stirring, about 8 minutes.
Return the meat mixture to the pot. Add the cognac and scrape up the browned bits. Cook, stirring, until the liquid is absorbed, about 5 minutes. Add 2 cups water and the reserved mushroom soaking liquid. Crush the tomatoes with your hands over the pot and add with their juices. Simmer over medium-low heat, stirring occasionally, until thickened, about 2 hours, 30 minutes. Cool completely.
Meanwhile, make the bechamel: Melt the butter in a wide saucepan over medium heat. Add the flour, nutmeg, 2 teaspoons salt and 1/2 teaspoon pepper; cook, whisking, 2 to 3 minutes. Gradually whisk in the milk and cook, whisking, until thickened, 5 to 10 minutes. Reduce the heat to low and cook, whisking, 10 more minutes.
Assemble the lasagna: Preheat the oven to 400 degrees F. Bring a few inches of salted water to a boil in a wide saucepan. Fill a large bowl with ice water. Line a baking sheet with a damp kitchen towel. Working in batches, boil the pasta until partially cooked, about 4 minutes, then remove with a skimmer and transfer to the ice water to cool. Remove the pasta sheets, shake off the excess water and arrange in a single layer on the towel. Cover the pasta with another damp towel.
Brush the bottom and sides of a deep 11-by-13-inch baking dish with the butter. Spread 1/2 cup ragu in the pan. Cover with a layer of pasta, then sprinkle with 1/3 cup parmesan and 2/3 cup mozzarella. Spread 1 1/2 cups ragu over the cheese and top with 1 cup bechamel. Repeat with 3 more layers each of pasta, parmesan, mozzarella, ragu and bechamel. Top with a layer of pasta, then cover with the remaining bechamel. Sprinkle with the remaining parmesan and mozzarella. Tuck the edges of the pasta into the baking dish with a knife.
Place the lasagna on a baking sheet to catch any drips; bake until bubbly and golden, about 40 minutes. Let rest 30 minutes before slicing and serving.