Boil noodles according to package directions and drain.
Preheat oven to 350 degrees. Lightly grease an 8" x 8" baking pan. Add squash to cold, salted water in a pot and bring to a boil. Allow to boil for about 20 minutes, until fork-tender. Drain, reserving about 1/2 cup of the boiling liquid.
Add half the squash and half of the reserved boiling water to a blender. Puree until smooth and creamy. Pour into a bowl. Repeat with remaining half of the squash and the water.
Mix elbow noodles with squash and olive oil. Sprinkle in cheese, salt, nutmeg, and pepper. Mix well. Pour mixture into prepared baking pan and cover well with foil. Bake for about 20 minutes, remove foil and bake for an additional 5 to 10 minutes, until cooked through.