PreparationFor soup: Preheat oven to 400°F. Heat oil in a large pot over medium-high heat. Season chicken with salt and pepper. Cook skin side down until skin is golden, 8-10 minutes; transfer to a plate. Carefully drain all but 2 tablespoons fat from pot. Add carrots, celery root, onion, and leek; stir frequently until softened and beginning to brown, 8-10 minutes. Add butter; stir until melted. Add flour and stir constantly until well blended, about 2 minutes. Add wine and 1/2 cup water. Bring to a boil; reduce heat and simmer until liquid is reduced by half, 5-8 minutes. Add chicken, broth, thyme, rosemary, and bay leaves. Bring to a boil; reduce heat. Simmer until chicken is fork-tender, 35-40 minutes. Transfer chicken to a plate. Let cool slightly; shred meat, discarding skin and bones. Return shredded chicken to soup. Add 1 1/2 teaspoons vinegar. Season with salt, pepper, and more vinegar, if desired. DO AHEAD: Soup can be made 2 days ahead. Let cool slightly; chill until cold. Cover and keep chilled. Reheat before serving. For biscuits: Whisk flour, salt, and baking powder in a large bowl. Add butter. Using your fingertips, blend until pea-size lumps form. Add buttermilk, crème fraîche, and chives. Using a fork, mix until just combined. Gather mixture into a ball and knead in bowl just until a shaggy dough forms, 3 or 4 times. Transfer dough to a floured work surface and pat into a 3/4"- to 1"-thick round. Cut out rounds with a 2" biscuit cutter or small glass. Gather dough and repeat patting and cutting. Transfer to a parchment-lined rimmed baking sheet; brush with egg. Bake biscuits until golden brown, 25-30 minutes. Serve warm or at room temperature. DO AHEAD: Biscuits can be made 8 hours ahead. Let cool completely and store airtight at room temperature. Divide soup among bowls and top with biscuits.
PreparationFor soup: Preheat oven to 400°F. Heat oil in a large pot over medium-high heat. Season chicken with salt and pepper. Cook skin side down until skin is golden, 8-10 minutes; transfer to a plate. Carefully drain all but 2 tablespoons fat from pot. Add carrots, celery root, onion, and leek; stir frequently until softened and beginning to brown, 8-10 minutes. Add butter; stir until melted. Add flour and stir constantly until well blended, about 2 minutes. Add wine and 1/2 cup water. Bring to a boil; reduce heat and simmer until liquid is reduced by half, 5-8 minutes. Add chicken, broth, thyme, rosemary, and bay leaves. Bring to a boil; reduce heat. Simmer until chicken is fork-tender, 35-40 minutes. Transfer chicken to a plate. Let cool slightly; shred meat, discarding skin and bones. Return shredded chicken to soup. Add 1 1/2 teaspoons vinegar. Season with salt, pepper, and more vinegar, if desired. DO AHEAD: Soup can be made 2 days ahead. Let cool slightly; chill until cold. Cover and keep chilled. Reheat before serving. For biscuits: Whisk flour, salt, and baking powder in a large bowl. Add butter. Using your fingertips, blend until pea-size lumps form. Add buttermilk, crème fraîche, and chives. Using a fork, mix until just combined. Gather mixture into a ball and knead in bowl just until a shaggy dough forms, 3 or 4 times. Transfer dough to a floured work surface and pat into a 3/4"- to 1"-thick round. Cut out rounds with a 2" biscuit cutter or small glass. Gather dough and repeat patting and cutting. Transfer to a parchment-lined rimmed baking sheet; brush with egg. Bake biscuits until golden brown, 25-30 minutes. Serve warm or at room temperature. DO AHEAD: Biscuits can be made 8 hours ahead. Let cool completely and store airtight at room temperature. Divide soup among bowls and top with biscuits.