Chicken Mushroom (and Spinach) Lasagna

Preheat oven to 400° and cook seasoned chicken breast for 30 minutes. Once cooked, shred chicken, put into a large bowl and set aside. Now bring the oven to 425° with rack in middle. Cook mushrooms, spinach, garlic, 1/4 teaspoon salt, and 1/8 teaspoon pepper in olive oil and add 1 tablespoon butter in a saucepan over medium heat, stirring occasionally, until mushrooms are softened, about 5 minutes. Add wine and simmer briskly 2 minutes. Transfer mushroom mixture to a large bowl and stir in chicken. Bring milk to barely a simmer in a medium saucepan. In a different pan, Melt remaining 4 tablespoons butter medium-low heat. Add flour and cook roux, whisking constantly, 3 minutes. Add hot milk in a slow stream, whisking constantly. Add thyme, 3/4 teaspoon salt, and 1/2 teaspoon pepper and simmer, whisking occasionally, until thickened, 5 to 6 minutes. Remove from heat and reserve 1 cup sauce. Stir Parmesan into sauce remaining in pan, then stir into mushroom filling. Pour half of reserved plain sauce into baking pan, spreading evenly to coat bottom. Add 3 lasagna noodles, overlapping slightly, and one third of mushroom filling, spreading evenly, then sprinkle one fourth of Gruyère over top. Repeat 2 times. Top with remaining 3 lasagna noodles and remaining plain sauce, spreading evenly. Sprinkle with remaining Gruyère. Cover with foil, tenting slightly to prevent foil from touching top of lasagna but sealing all around edge, and bake 30 minutes. Remove foil and bake until cheese is golden, about 15 minutes more. Let lasagna stand at least 10 minutes before serving.