Make the sauce: Soak the dried porcini in 3 cups hot water, 30 minutes. Drain, reserving 2 cups of the soaking liquid. Roughly chop the porcini.
Heat 1 tablespoon each butter and olive oil in a large pot or Dutch oven over medium-high heat. Add half of the button and shiitake mushrooms and cook, undisturbed, until slightly browned on the bottom, about 2 minutes. Continue to cook, stirring, until soft, 2 to 3 more minutes; transfer to a bowl. Repeat with another 1 tablespoon each butter and olive oil and the remaining button and shiitake mushrooms. Transfer to the bowl and add 1/2 teaspoon salt.
Reduce the heat to medium and add the remaining 1 tablespoon each butter and olive oil to the skillet. Add the onion and porcini and cook, stirring, until the onion is soft, about 4 minutes. Stir in the garlic and red pepper flakes. Add the tomato paste and cook, stirring, until darkened, 1 to 2 minutes. Add the reserved 2 cups porcini liquid, the crushed tomatoes, button and shiitake mushrooms and 1/2 teaspoon salt. Bring to a simmer and cook, stirring occasionally, until thickened, about 30 minutes. Remove from the heat and stir in the parsley; season with salt.
Prepare the lasagna: Bring 7 cups water to a boil in a large saucepan; transfer to a large heatproof bowl. Meanwhile, bring 3 cups water to a boil in a separate large saucepan. Gradually whisk the polenta and 1 1/2 teaspoons salt into the bowl of hot water. Cover the bowl tightly with foil and set over the saucepan of boiling water. Reduce the heat so the water is simmering; cook the polenta until the liquid is absorbed, about 15 minutes. Remove from the heat, uncover and vigorously whisk in the parmesan, eggs and nutmeg.
Preheat the oven to 350 degrees F. Lightly brush a 3-quart baking dish with olive oil. Spread one-third of the polenta in the dish and top with half of the mushroom sauce. Sprinkle with 1 cup mozzarella. Repeat the layers (polenta, sauce, mozzarella), then top with the remaining polenta. (If the polenta gets too thick, whisk in a little hot water.)
Loosely cover the dish with foil and bake until bubbly, 50 minutes to 1 hour. Turn on the broiler. Uncover the lasagna and top with the remaining 1 cup mozzarella; broil until melted, 2 to 3 minutes. Sprinkle with parsley. Let stand 15 minutes before serving.