Preheat oven to 375 degrees. Combine squash, stock, and milk in a medium saucepan; bring to a boil over medium-high heat. Reduce heat to medium; simmer until squash is tender when pierced with a fork, about 20 minutes. Remove from heat. Mash contents of saucepan; stir in nutmeg, cayenne, and salt, and season with black pepper. Stir to combine.
Meanwhile, bring a large pot of water to a boil. Add noodles; cook until al dente according to package instructions, about 8 minutes. Drain, and transfer to a large bowl; stir in squash mixture, cheddar, ricotta, and 2 tablespoons Parmesan.
Lightly coat a 9-inch square baking dish (4 inches deep) with cooking spray. Transfer noodle mixture to dish. In a small bowl, combine breadcrumbs, remaining 2 tablespoons Parmesan, and oil; sprinkle evenly over noodle mixture.
Cover with foil, and bake 20 minutes. Remove foil, and continue baking until lightly browned and crisp on top, 30 to 40 minutes more. Serve immediately.