Heat the butter in a large wide pot over medium heat. Add the garlic, sage sprigs, shallots, 1 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the shallots soften, about 5 minutes. Increase the heat to medium-high, add the carrots and continue to cook, stirring occasionally, until the butter begins to brown and the carrots are tender but not fully cooked, about 12 minutes. Add the cider and 2 1/2 cups water and bring to a boil. Reduce the heat to a simmer and continue to cook, stirring occasionally until the carrots are very soft, about 20 minutes. Remove from the heat, discard the sage sprigs and cool the soup slightly.
Meanwhile, heat the oil in a small skillet over medium-high heat. Working in batches, add the sage leaves and cook until bright green and crisp, 20 to 30 seconds. Transfer to a paper-towel-lined plate, sprinkle with salt and set aside.
Puree the soup with an immersion blender in the pot or, working in batches, puree in a blender until smooth, taking care not to fill the carafe more than half full per batch. Reheat the soup over medium heat. Stir in the vinegar and adjust the seasoning with additional salt and pepper. Ladle the soup into bowls and top with a small spoonful of creme fraiche and a crispy sage leaf.
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