Roast squash: Preheat oven to 400 degrees F. Roast squash, cut sides down, in an oiled roasting pan in middle of oven until tender, about 1 hour. Set aside to cool. When cool enough to handle, scrape flesh from skin.
Fry sage leaves while squash roasts: Heat vegetable oil in a small, deep saucepan until it registers 365 degrees on a deep-fry thermometer. Fry sage leaves in 3 batches until crisp, about 3 to 5 seconds. Transfer leaves with a slotted spoon to paper towels to drain.
Cook pancetta and make soup: Cook pancetta in a 4-quart heavy pot over moderate heat, stirring, until browned. Transfer pancetta with slotted spoon to paper towels to drain.
Add olive oil to pancetta fat remaining in pot, then cook onion, stirring, until softened. Stir in garlic and chopped sage and cook, stirring, until fragrant, about 1 minute. Add squash, broth, and water and simmer 20 minutes to blend flavors.
Puree soup in batches in a blender, transferring to a bowl. (Use caution when blending hot liquids.) Return soup to pot and reheat. If necessary, thin to desired consistency with water. Stir in vinegar and salt and pepper, to taste.
Serve sprinkled with pancetta and fried sage leaves.
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