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Preheat oven to 425 degrees F.
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Toss delicata with oil, salt, and spices. Arrange on a large rimmed baking sheet with plenty of room between slices. Roast 15 minutes, flip squash and roast 10 - 15 minutes longer, until squash is a deep golden hue and edges has crisped up.
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Meanwhile, bring a large pot of water to a boil. Add sea salt and spaghetti and cook until pasta is just al dente. Drain pasta, reserving 1/2 cup pasta water.
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In a large Dutch oven or 3- or 4-quart saucepan over medium heat, heat olive oil. Add speck, cooking just until crispy and fragrant, 2 - 3 minutes. Remove from pot and set on paper towels to cool, then rough chop.
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In the same pot, heat the butter over medium-low heat. When melted, stir in garlic. Cook until fragrant, 3 - 4 minutes. Stir in cream, and then the Parmesan. When Parmesan has melted, remove pot from heat. Stir in 1/4 teaspoon sea salt (or add to taste) and pepper. Fold in spaghetti. Add reserved pasta water 1 tablespoon at a time to loosen sauce. Finally, toss with arugula and parsley.
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Serve immediately with roasted delicate squash tucked throughout.