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Cook Squash: Preheat oven to 400 degrees. Cut off stem of the squash and slice vertically. Hollow out the seeds with a spoon and discard. Rub the inside of the squash with some olive oil, just until coated, and season with salt and pepper. Place face up on a cookie sheet and bake for 45 minutes or until fork tender.
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Cook Pasta: Meanwhile, cook pasta until al dente in salted water.
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Prepare Cheese Sauce: melt butter in a pan on low heat. Once melted, whisk in flour to form a roux. Whisk in milk a little at a time, stirring constantly. Allow the sauce to thicken over low heat, continuously stirring. (Be careful not to boil.)
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Once all the milk is incorporated, remove from heat. Add nutmeg, salt, pepper, cheddar cheese, and half of the parmesan cheese. Stir until cheese is melted.
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When squash is cooked through, allow to cool until you can handle (or use oven mitts). Scrape out the inside of the squash into a bowl or pot. Add half of the cheese sauce and stir well.
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Add the drained pasta to your squash-cheese mixture. Stir until all pasta is covered in sauce and add to a greased baking dish.
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Pour remaining cheese sauce over the pasta. Top with remaining parmesan, mozzarella, and finish with breadcrumbs.
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Bake at 350 degrees for 20 minutes, or until breadcrumbs just begins to brown.