Heat olive oil or butter in a large stockpot over medium heat until shimmering and fragrant.
Add the chicken pieces and sear until golden on both sides, about 5 minutes per side. Use tongs to turn them carefully, and listen for a gentle sizzle.
Remove the chicken and set aside; in the same pot, add chopped onions, carrots, and celery. Sauté until vegetables are fragrant and slightly caramelized, about 10 minutes, stirring occasionally.
Add the minced garlic and thyme sprig to the vegetables, cooking for another minute until fragrant.
Pour in water or broth, scraping the bottom of the pot to loosen any browned bits, and bring to a gentle boil.
Return the seared chicken to the pot, ensuring it’s submerged in the liquid. Reduce heat to low, cover, and let simmer gently for about 1 hour, skimming foam as needed.
Check the chicken for tenderness; it should be easy to shred with a fork. Remove the chicken and set aside to cool slightly.
Remove the thyme sprig and discard. Shred the chicken into bite-sized pieces, then return it to the pot.
Squeeze fresh lemon juice into the soup, stirring well to brighten the flavors. Season with salt and pepper to taste.
Let the soup rest for 10 minutes so flavors meld, then sprinkle chopped parsley on top before serving.
Serve hot with crusty bread or your favorite side, and enjoy the cozy, nourishing flavors.