Preheat the oven to 400°F (200°C). Melt 3 tablespoons butter in a large saucepan, then add minced garlic and cook for 30 seconds until fragrant. Add spinach in batches, wilting each batch before adding more, then stir in 1 cup heavy cream, nutmeg, salt, and pepper; simmer for 15 minutes until thick and reduced.

In a food processor, combine cottage cheese, egg, and 6 tablespoons parsley, then process for about 5 seconds until the cottage cheese has a ricotta-like texture.

Combine the cooked spinach and cottage cheese mixture in a large bowl. Separately, soak lasagna noodles in warm water in a baking dish for 15 minutes until lightly softened, then dry them on a clean kitchen towel.

In the same pot, melt 3 more tablespoons butter and cook finely chopped mushrooms for 10 minutes until their liquid evaporates. Add shallots and thyme, cooking for 2 minutes until fragrant, then stir in soy sauce, lemon juice, and the remaining heavy cream, simmering for 3 minutes until lightly thickened; season with salt and pepper and transfer to another bowl.

Wipe out the pot, melt the remaining 2 tablespoons butter, and whisk in flour until light golden blond. Slowly add milk while whisking constantly, bring to a simmer, then remove from heat and stir in 2/3 of the mozzarella and Parmesan; season with salt and pepper.

Grease an 8- by 13-inch baking dish. Layer 1 cup cheese sauce, three lasagna sheets, half of the mushroom mixture, three more lasagna sheets, half of the spinach/cottage cheese mixture, and three more sheets of noodle. Repeat these layers with the remaining mushroom mixture, spinach mixture, and noodles, ending with a final noodle layer. Pour the remaining cheese sauce over the top and sprinkle with the remaining 1/3 grated mozzarella.

Bake the lasagna for 20 minutes until bubbling, then broil for about 5 minutes until the top is lightly browned. Let cool for 10 minutes before slicing, serving, and garnishing with chopped parsley.