For the meat sauce: Heat 2 tablespoons of the oil in a large Dutch oven over high heat. Add the mushrooms, thyme and 1/4 teaspoon salt and cook, stirring occasionally, until the mushrooms have released their liquid and begin to brown, 5 to 7 minutes. Transfer the mushrooms to a plate and reserve.
Heat 2 tablespoons of the oil in the Dutch oven over high heat. Add the ground beef and 1 teaspoon salt and cook, breaking the meat up with a wooden spoon, until it browns, about 5 minutes. Add the remaining tablespoon of oil, onions, rosemary and oregano. Reduce the heat to medium and cook until the onions soften, about 8 minutes. Add the garlic and cook, stirring, until it is fragrant, about 2 minutes. Add the wine, scrape up any browned bits on the bottom of the pan and reduce until dry, about 10 minutes. Add the tomatoes, reserved mushrooms, 1 teaspoon salt and 1/2 teaspoon pepper and bring to a boil over high heat. Reduce the heat to medium and cook until the sauce is deep red and has reduced to 6 cups, about 40 minutes.
For the bechamel: Meanwhile, preheat the oven to 350 degrees F. Melt the butter in a large saucepan over medium-high heat, sprinkle in the flour and whisk to form a paste. Add the milk while continuing to whisk, raise the heat to high and bring to a boil. Reduce the heat to medium-low and simmer, stirring occasionally, until the sauce is thick and creamy and has reduced to6 cups, about 10 minutes. Add the nutmeg and 1 teaspoon salt.
To assemble: Butter the bottom of a 9-by-13-inch lasagna dish and cover with 3 lasagna sheets. Top with 1 1/2 cups each of the meat sauce and the bechamel and sprinkle with 1/4 cup of the Parmesan. Repeat with the remaining pasta, meat sauce, bechamel and Parmesan for a total of 4 layers, ending with 1/2 cup of Parmesan. Cover with foil and bake until bubbly, about 40 minutes. Uncover and bake until the top becomes golden, about 10 minutes more. Let stand 20 minutes before serving.