Place the chicken bones and carcass in a large stockpot and cover with cold water. Bring to a gentle simmer over medium heat, allowing the water to bubble softly.
Skim off any foam that rises to the surface during the first 15 minutes to keep the broth clear and clean in flavor.
Add the chopped onions, carrots, and celery to the pot. Continue simmering gently for at least two hours, uncovered, allowing the flavors to deepen and the broth to become rich.
Occasionally check the broth, adding more water if necessary to keep the ingredients submerged. The broth should develop a fragrant aroma and a slightly gelatinous texture when done.
Strain the broth through a fine sieve or chinois into a clean pot or bowl, discarding the solids. You should be left with a clear, golden liquid.
Season the broth with salt and pepper to taste, adjusting slowly to build a balanced depth of flavor.
Add the shredded cooked chicken to the hot broth, and stir in the chopped fresh thyme for a herbal lift.
Squeeze fresh lemon juice into the soup for brightness, then stir in chopped parsley just before serving.
Drizzle a good-quality olive oil over each bowl just before serving to add richness and shine.
Serve the soup hot, garnished with additional herbs and a slice of lemon if desired, enjoying the comforting aroma and tender ingredients.