Preheat the oven to 375 degrees F (190 degrees C). Grease a 9x13-inch baking dish with cooking spray.
Place sausage in a skillet over medium heat and cook until browned and crumbly, about 10 minutes. Drain and set aside.
To make béchamel sauce, melt butter in a saucepan over medium heat and whisk in flour. Cook until light and foamy, about 2 minutes, whisking constantly; whisk in milk, a little at a time. Whisk in salt and nutmeg. Bring sauce to a bare simmer, whisking constantly until thickened, about 10 minutes. Remove sauce from heat and set aside.
Beat egg in a medium bowl; stir in ricotta cheese and parsley until thoroughly combined. Set aside.
To assemble lasagna, pour about 1 1/2 cups marinara sauce into the bottom of the prepared baking dish. Top sauce with 4 lasagna noodles, overlapping if necessary. Layer 1/3 of the ricotta mixture, followed by 1/3 of the sausage, 1/3 cup mushrooms, 1/4 of the béchamel sauce, 1 cup mozzarella cheese, 1/8 cup Parmesan cheese, and 1 more cup marinara sauce over noodles. Repeat the layers twice more, ending with a final layer of noodles topped with remaining marinara sauce, béchamel sauce, mozzarella, and Parmesan cheese. Cover the dish with foil.
Bake in the preheated oven until bubbly and noodles are tender, about 1 hour. Remove the foil and bake until the top cheese layer is browned, about 5 more minutes. Allow lasagna to stand for 10 minutes before cutting.