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Cut off neck of squash and reserve bulbous end, wrapped and chilled, for later use. Peel neck and cut into roughly 1-inch cubes (3 1/2 to 4 cups; 1 lb). Process in a food processor until very finely chopped. Combine with water (2 cups) and 1/2 tsp salt in a heavy medium saucepan and briskly simmer, uncovered, stirring once or twice, until tender, 10 to 15 minutes.
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Chop 1 Tbsp sage. While squash is simmering, cook onion in olive oil with 1/4 tsp salt in a 12-inch heavy skillet over medium-high heat, stirring occasionally, until lightly browned, 8 to 10 minutes. Add garlic and chopped sage and cook, stirring, 1 minute. Add squash and simmer, stirring, 2 minutes. Stir in cheese and season with salt and pepper.
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Cook rotini in a pasta pot of well-salted boiling water until al dente.
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While pasta is cooking, pat sage leaves dry, if necessary, with paper towels. Heat vegetable oil in an 8- to 9-inch skillet over high heat until it shimmers. Test frying temperature by dipping tip of a leaf in oil. If it bubbles briskly, it’s ready. Add a small handful of leaves (they will make quite a commotion) and fry, stirring with a slotted spoon, until crisp but still green, 10 to 30 seconds. Transfer to paper towels to drain and fry remaining leaves in 3 or 4 batches. Season with salt.
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Reserve 1 cup pasta-cooking water, then drain pasta and add to sauce. Cook over medium heat, stirring, until well coated, about 1 minute, thinning if necessary with some of reserved cooking water.
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Transfer to a large bowl and serve fried sage leaves on the side (each person can top their own serving with the leaves).
