Adjust the oven racks to the upper-middle and middle positions and preheat the oven to 400°F.

Heat 3 tablespoons of the butter in a large saucepan over medium-high heat until the foaming subsides. Add the garlic and cook, stirring, until fragrant, about 30 seconds.

Add the spinach in batches, allowing each batch to wilt before adding the next. Once all the spinach has been added, add 1 cup of the heavy cream and bring to a boil.

Reduce to a strong simmer and cook, stirring frequently, until thickened and reduced, about 15 minutes. Add the nutmeg, season to taste with salt and pepper, and remove from the heat.

While the cream is reducing, combine the cottage cheese, egg, and 3 tablespoons of the parsley in the bowl of a food processor and process until combined and the cottage cheese is the texture of ricotta, about 5 seconds. Transfer to a large bowl, add the cooked spinach, and mix well.

Meanwhile, place the lasagna noodles in a 9- by 13-inch baking dish and cover with warm water. Allow to soak, agitating the noodles occasionally to prevent sticking, until lightly softened, about 15 minutes. Drain and arrange in a single layer on paper towels or a clean kitchen towel to dry.

While the noodles soak, wash the spinach pan and return it to medium-high heat. Add 3 more tablespoons butter and heat until melted. Add the mushrooms and cook, stirring occasionally, until the liquid evaporates and the mushrooms start to sizzle, about 10 minutes.

Add the shallots and thyme and cook, stirring frequently, until the shallots are softened, about 2 minutes. Add the soy sauce and lemon juice and stir to combine. Add the remaining 1 cup heavy cream, bring to a simmer, and cook until lightly thickened, about 3 minutes. Season to taste with salt and pepper and transfer to a bowl.

Wipe out the saucepan, return to medium-high heat, add the remaining 2 tablespoons butter, and heat until melted. Add the flour and cook, whisking constantly, until light golden blond.

Slowly pour in the milk, whisking constantly. Bring to a simmer, then remove from the heat and stir in two-thirds of the mozzarella and the Parmesan, then season to taste with salt and pepper.

To assemble the lasagna, dry the baking dish and grease with butter. Spread 1 cup of the cheese sauce in the bottom of the dish.

Lay 3 noodles on top of it, spacing them evenly (the noodles will not quite touch each other; this is fine). Top the noodles evenly with half of the mushroom mixture and then with another 3 noodles.

Top with half of the spinach/cottage cheese mixture and another 3 noodles. Repeat the layers with the remaining mushroom mixture, spinach mixture, and noodles, ending with a layer of noodles.

Pour the remaining cheese sauce over the top and spread it evenly. Sprinkle the remaining mozzarella evenly over the top.

Place a foil-lined baking sheet on the lower oven rack to catch any drips, then place the lasagna on the rack above it. Bake until the lasagna is bubbling around the edges, about 20 minutes.

Turn on broiler on and broil until the top is lightly browned, about 5 minutes (if you have an under-oven broiler, transfer the lasagna to the broiler after baking).

Let cool for 10 minutes, then slice and serve.