Preheat oven to 425.

Combine squash, ½ tablespoon oil, sage, garlic, ½ teaspoon salt, and black pepper in a large bowl; toss to coat.

Arrange squash mixture on a baking sheet coated with cooking spray. Bake for 30 minutes or until squash is tender.

Heat remaining oil in a large Dutch oven over medium-high heat. Add onion, and sauté for 4 minutes. Reduce heat to medium-low; continue cooking for 20 minutes or until golden brown, stirring frequently. Place onion in a bowl.

Steam spinach in microwave safe steamer for 3 minutes.  Cool, squeeze excess liquid from spinach.  Add to onions.

Heat 2 ½ cups milk, and thyme in a medium saucepan over medium-high heat. Bring to a boil; remove from heat. Let stand for 10 minutes. Discard thyme. Return pan to medium heat.

Combine remaining ½ cup milk and flour in a small bowl. Add to pan, stirring with a whisk until blended. Bring to a boil; reduce heat, and simmer for 5 minutes or until thickened, stirring constantly. Remove from heat; stir in  teaspoon salt, cheese, cayenne, and nutmeg.

Spread ½ cup milk mixture in bottom of a 8 x 8-inch glass or ceramic baking dish coated with cooking spray. Arrange noodles to fit over milk mixture; top with half of squash mixture, half of spinach mixture, and ¾ cup milk mixture. Repeat layers. Spread remaining milk mixture over noodles. Top with ½ cup cheese.

Bake for 30 minutes, and remove from oven. Let lasagna sit 10 minutes prior to serving.