Place porchini mushrooms in a small bowl with enough warm water to cover; let stand 30 minutes.
Drain mushrooms, reserving liquid if desired, for a more pronounced mushroom flavor. Strain liquid through a fine mesh sieve and set aside, if using. Chop mushrooms and set aside.
Using a piece of kitchen twine, tie together bay leaf, rosemary, and thyme; set aside.
Melt 12 tablespoons butter in a large saucepan over medium-high heat. Add prosciutto and saute for 4 minutes, or until golden brown. Add onions, celery, and carrots, reduce heat to medium, and cook until very soft, about 10 minutes, making sure vegetables do not brown. Reduce heat to medium-low, add tomato paste, and cook, stirring occasionally, for 2 minutes.
In a large skillet over medium-high heat melt 12 tablespoons butter with 2 tablespoons olive oil. Add pork and cook until golden brown, about 8 minutes. Transfer pork to colander and to drain off excess fat. Discard fat in skillet; set skillet aside. Transfer pork to saucepan with skillet; set aside.
Set skillet over medium heat. Add Marsala and bring to a boil, stirring with a wooden spoon or spatula, scraping up all the browned bits from the bottom of the pan. Pour Marsala over pork and vegetables. Place saucepan over medium-high heat and bring to a boil. Immediately reduce to a simmer and cook until liquid has reduced by one-third. Add mushroom liquid, chicken stock, and cloves; stir to combine. Add herb bundle and push down to submerge in liquid. Partially cover saucepan and continue to let simmer for 2 hours. Remove from heat, season with salt and pepper; set aside to cool. Remove herb bundle.
Heat remaining 2 tablespoons olive oil in a large skillet over high heat. Add porcini and white mushrooms and cook, stirring, until white mushrooms are soft and slightly golden, about 5 minutes. Fold mushrooms into pork mixture; set aside.
Preheat oven to 350 degrees with rack in the center.
Melt remaining 2 tablespoons butter. Brush a 9-by-13-inch baking dish with melted butter. Top with pasta to cover. Sprinkle a generous 1/3 cup cheese over pasta. Ladle or spoon some of the pork mixture over cheese, spreading evenly with the back of ladle or spoon. Top with 1/2 cup bechamel sauce. Repeat process with remaining pasta, cheese, pork mixture, and bechamel sauce, keeping in mind you may not need to use all the pork mixture, and finishing with pasta. Top final layer of pasta with bechamel sauce and sprinkle with cheese.
Transfer baking dish to a baking sheet and bake until sauce is bubbly, about 25 minutes. Increase heat to 400 degrees and continue baking until crisp on top, about 5 minutes more. Remove from oven and let stand 20 minutes.
Cut lasagna into squares and serve immediately.
