Butternut Squash Lasagna

Name: Butternut Squash Lasagna
Ingredients: 1 butternut squash, halved and seeded; 1 tablespoon olive oil, divided, or as needed; 1 (8 ounce) package sliced fresh mushrooms; 1 teaspoon minced garlic; 1 onion, chopped; 1 (10 ounce) package frozen chopped spinach, thawed and drained; 7 cups milk, divided; 0.5 cup cornstarch; 2 cups grated Parmesan cheese, divided; 1 tablespoon honey; 0.25 teaspoon ground nutmeg; 4 fresh sage leaves, shredded, or to taste; salt and ground black pepper to taste; 1 (16 ounce) package lasagna noodles; 0.25 teaspoon minced fresh ginger; 2 cups shredded mozzarella cheese