Preheat oven to 400 degrees. Coat a 9x13 inch baking dish with cooking spray.
Heat olive oil in a large skillet over medium heat. Add the diced onion; season with salt and pepper. Cook for 6-8 minutes until onions are browned and starting to caramelize. Stir in the rosemary.
Meanwhile, cook the pasta in salted boiling water. Drain and reserve 1 and 1/2 cups of the pasta cooking water.
Stir the squash puree and reserved pasta water into the onion mixture; simmer 3 minutes. Toss squash mixture, 1 cup of white cheddar cheese and 1/2 cup of parmesan cheese with the pasta. Transfer to prepared dish.
Sprinkle the remaining 1/2 cup of white cheddar cheese over the top of the pasta. Mix the breadcrumbs with the butter and evenly sprinkle the breadcrumbs over the cheese.
Bake for 20-25 minutes or until pasta is bubbling and breadcrumbs are golden brown.