Preparation

• Adjust an oven rack to the upper-middle position and preheat oven to 400°F (200°C). Melt 3 tablespoons butter in a large saucepan over medium-high heat until foaming subsides. Add garlic and cook, stirring constantly, until fragrant, about 30 seconds. Add spinach in batches, allowing previous batch to wilt before adding next. Once all spinach is added, add 1 cup heavy cream. Bring to a boil, and reduce to a strong simmer. Cook, stirring frequently, until thick and reduced, about 15 minutes. Add nutmeg, and season to taste with salt and pepper.
• While spinach is cooking, combine cottage cheese, egg, and 6 tablespoons parsley in the bowl of a food processor. Process until combined and cottage cheese is broken up to the texture of ricotta cheese, about 5 seconds.
• Combine cooked spinach and cottage cheese mixture in a large bowl. Meanwhile, place lasagna noodles in a 8- by 13-inch baking dish and cover with warm water. Allow to soak, agitating occasionally to prevent sticking, until lightly softened, about 15 minutes. Transfer in a single layer to a clean kitchen towel to dry.
• While noodles soak, wipe out spinach pot and return to medium-high heat. Add 3 more tablespoons butter and heat until melted. Add mushrooms and cook, stirring occasionally, until liquid evaporates and mushrooms start to sizzle, about 10 minutes. Add shallots and thyme and cook, stirring frequently, until fragrant and softened, about 2 minutes. Add soy sauce and lemon juice and stir to combine. Add remaining heavy cream. Bring to a simmer, and cook until lightly thickened, about 3 minutes. Season to taste with salt and pepper. Transfer mixture to another bowl.
• Wipe out pot and return to medium-high heat. Add remaining 2 tablespoons butter and heat until melted. Add flour and cook, stirring constantly with a whisk, until light golden blond. Slowly pour in milk, whisking constantly. Bring to a simmer, then remove from heat. Stir in ⅔ of mozzarella and Parmesan, then season to taste with salt and pepper.
• To assemble lasagna, dry the 8- by 13-inch baking dish and grease with butter. Add 1 cup cheese sauce to bottom of dish. Lay three sheets of noodle on top of it, spacing them evenly across the bottom of the dish (there will be about ¼-inch gap between each sheet of noodle and the edge of the pan. This is fine). Top noodle evenly with half of mushroom mixture, then top with another three sheets. Top with half of spinach/cottage cheese mixture, then top with another three sheets of noodle. Repeat layers with remaining mushroom mixture, spinach mixture, and noodle, ending with a layer of noodle. Pour remaining cheese sauce over top and spread evenly. Sprinkle remaining ⅓ of grated mozzarella evenly over top.
• Bake until lasagna is bubbling around the edges, about 20 minutes. Switch broiler on and broil until top is lightly browned, about 5 minutes longer (if you have an under-oven broiler, you'll have to transfer it to the broiler after baking). Let cool for 10 minutes, then slice and serve, garnish with chopped parsley.