Preheat oven to 425 degrees. Grate about half of the squash until you have 5 cups cut remaining squash into ½-inch cubes (about 4 cups). On a rimmed baking sheet, toss squash cubes with 1 tablespoon oil; season with salt and pepper. Spread in a single layer and roast until tender and browned, about 25 minutes.
Meanwhile, cook pasta in a large pot of boiling salted water according to package directions. Drain, reserving 1 ½ cups pasta water. Heat a large skillet over medium-high. Add remaining 2 tablespoons oil and shredded squash; cook, stirring occasionally, until caramelized in spots, 10 minutes. Transfer to a blender with reserved pasta water and half of cheese; blend until smooth.
Return blended sauce to skillet; stir in pasta and roasted squash. Season with salt and pepper. Serve immediately, topped with arugula, hazelnuts, small dollops of remaining cheese, and a drizzle of oil.