Heat the olive oil and butter in a large pot over medium heat. Add the onion and sauté about 5 minutes until softened.
Add the garlic, carrot, potatoes, celery root, and tomato. Cook for 2–3 minutes, stirring.
Add the chicken, water, bay leaf, salt, and pepper. Bring to a boil; add vinegar if using.
Reduce to a gentle simmer for about 40 minutes, skimming foam as needed. Check the thickest part of the chicken reaches at least 165°F (74°C), without touching bone, and is tender enough to shred.
Remove the chicken. Discard the skin and bones, and shred the meat into generous pieces.
Remove the bay leaf and reserve about 1 cup broth. Blend the vegetables and remaining broth in batches until smooth. Allow hot soup to cool slightly and follow the blender manufacturer's hot-liquid safety instructions.
Return the blended soup to the pot. Add reserved broth gradually to adjust thickness, stir in shredded chicken, and season with salt, pepper, and lemon juice to taste.
Serve with croutons, parsley, and a drizzle of olive oil.