Heat the butter in a large pot over medium heat. Add the shallots, garlic, sage and 2 teaspoons salt and season with pepper. Cook, covered, stirring occasionally, until soft and fragrant, about 5 minutes. Increase heat to medium-high, add the squash and remaining 1 teaspoon salt, and continue to cook, stirring occasionally, until butter begins to brown and squash is tender, about 12 minutes. Add cider and 2 1/2 cups water and bring to a boil. Reduce heat and simmer, stirring occasionally, for 20 minutes. Remove the sage sprigs and set aside to cool slightly.
Heat the olive oil in a small skillet over medium high heat. Add the sage leaves, in two batches, and cook until bright green, 20 to 30 seconds. Transfer to a paper towel-lined plate. Sprinkle with salt. Set aside.
Puree the soup in a blender, (work in batches if needed) or with an immersion blender. Return soup to pot and reheat over medium heat. Stir in vinegar and transfer soup to warmed soup bowls. Serve in warmed soup bowls as is or garnished with creme fraeche or sage leaves. Serve with breadsticks.
When blending hot liquids: Remove liquid from the heat and allow to cool for at least 5 minutes. Transfer liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.
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