INGREDIENTS
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2 tablespoons butter
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1 cup mushrooms, such as oyster, mushroom, or maitake, roughly chopped
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2 pounds hard squash (approximately 1 large butternut, kabocha, acorn, or honeynut squash), peeled, seeded, and cut into 1-inch pieces
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1 medium onion, roughly chopped
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5 cloves garlic, minced, divided
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1 tablespoon fresh sage, minced
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Kosher salt
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Black pepper
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2 cups chicken or vegetable broth
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12 ounces dried pasta, such as fettuccine, spaghetti, or linguine
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12 cups water
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1/2 cup Pecorino or Parmesan cheese, grated, plus more for serving