Put the porcini mushrooms, marsala, and water into a small saucepan and bring to a boil, then take off the heat and scissor the softened mushrooms into smaller pieces.
Warm the oil in a heavy-based saucepan with a lid, and finely chop the onion, carrot, and celery either by hand or in a processor and scrape them into the pan.
Let the vegetables soften over a gentle heat for about 5 minutes or so, then stir in the thyme and salt.
Add the ground beef, breaking it up with a fork, and let it brown a little. Then stir in the tomato paste, canned tomatoes, and bay leaf.
Add the porcini mushrooms with their dark, gorgeously flavored liquid and bring the contents of the pan to a boil. Once the sauce is bubbling, put the lid on, turn the heat down to very low, and leave to simmer gently for about 45 minutes to 1 hour. If you're using a wide pan, the shorter time should be enough; if the sauce is piled up higher, you'll probably need the full hour.
While the sauce is simmering, make the polenta layers. First dampen your chosen pans 13x9x2 by letting some water from the cold tap splash them a bit.
Make up the polenta in a saucepan following the package instructions, but first dissolve a chicken bouillon cube in the specified amount of water, or use chicken broth instead of water.Stir as instructed with your wooden spoon, and when the polenta has thickened, add, beating as you go, the tablespoon of butter and 1 cup of Parmesan. Taste to see if you want any more seasoning. Once the polenta is thick and coming away from the sides of the saucepan, quickly divide it between the damp pans, spreading each one to an even as possible layer using a silicone spatula you've passed under a cold tap. It will set almost instantly. You can put aside these polenta layers and your meat sauce for now.
When you are ready to assemble the lasagne, preheat the oven to 400°F Using 1 polenta-lined pan as your dish, spoon half of the meat sauce over the polenta.
Deftly tip out (it's not hard) one of the other polenta layers and place this over the meat sauce in the pan you're working on and then add the last half of meat sauce.
Top with the final layer of polenta from the third pan, then sprinkle the remaining 1 cup of Parmesan cheese over the top.
Bake for 1 hour if your meat sauce was cold, and about 45 minutes if it was still warm. The cheese should have melted and become slightly golden, and the lasagne must be piping hot right through.
Make ahead note: The lasagne can be made and assembled 2 days ahead. Cover the pan tightly with plastic wrap or aluminum foil and refrigerate. Bake as directed in the recipe; however, the refrigerator-cold lasagne may need extra 10-20 minutes' cooking time, and check the lasagne is piping hot in the center.
Freeze note: The assembled lasagne can be frozen, wrapped in double layer of plastic wrap and a layer of foil, for up to 3 months. Thaw for 24 hours in the refrigerator then bake as above.