Preheat the oven to 450°F (230°C) and line a baking sheet with aluminum foil.

Place squash, cut-side up, on the prepared baking sheet. Cover with aluminum foil and bake until squash is tender, 45 minutes to 1 hour 15 minutes.

Heat 1 1/2 teaspoons olive oil in a skillet over medium heat. Cook and stir mushrooms and garlic in the hot oil until mushrooms are tender, 5 to 10 minutes. Remove from heat.

Heat remaining olive oil in a separate skillet over medium heat. Cook and stir onion until softened, about 5 minutes. Add spinach and cook until heated through, 3 to 5 minutes, and remove from heat.

In a 5-quart saucepan over medium heat, heat 6 cups of milk until simmering. Whisk remaining 1 cup of milk with cornstarch in a bowl until smooth, then stir into simmering milk. Cook and stir constantly until the mixture thickens into a sauce, about 5 minutes. Remove from heat.

Whisk Parmesan cheese (1.5 cups), honey, nutmeg, sage, and salt and pepper into the sauce until smooth. Cover and keep warm over low heat.

Bring a large pot of salted water to a boil. Cook lasagna noodles for about 8 minutes until firm, then drain.

Scrape squash flesh into a bowl and mash. Stir in ginger, salt, and pepper.

Reduce oven temperature to 375°F (190°C).

In a 9x13-inch baking dish, spoon 1 cup of white sauce. Arrange layers of lasagna noodles, squash mixture, spinach mixture, mushrooms, cheeses, and remaining sauce, following the given order. Cover with aluminum foil and place dish on a baking sheet.