1. Cook the mushrooms: Heat butter in a large skillet over medium-high heat and add the onion (reserve 2 tbsp for bechemel) and garlic; sauté for 2-3 minutes. Add in mushrooms and turn heat to high. Cook until tender, stirring often. about 10 mins. Add in thyme and marsala wine. Cook until most of the liquid evaporates, 15 minutes. Add salt and pepper to taste. Remove from heat.
2. Make the béchamel: Add 1/4 cup butter to a heavy saucepan over medium heat. Add onion and cook until softened. Stir in flour and cook for about 3 minutes, stirring the entire time. Whisk in the milk and bring to a simmer, stirring constantly, until it begins to thicken. Be sure to scrape the bottom and edges of the saucepan. Cook for 10-15 minutes, until thickened and the flour taste has cooked out.
3. Put the lasagna together: Heat the oven to 350F. Butter a casserole dish and place a thin layer of bechemel on the bottom. Top with a layer of noodles, then about a cup of the béchamel, a handful each of the mushrooms, chopped chicken, mozzarella and parmesan. Repeat layers two more times, ending with the mushroom béchamel layer topped with mozzarella and parmesan. Cover with foil and bake for 30 minutes. Remove the foil and bake an additional 10-1`5 minutes, until the top is lightly browned. Let cool slightly before serving.