Pour the chicken stock into a heavy-bottomed pot and bring it to a gentle simmer over medium heat, listening for soft bubbling and fragrant steam.
Add the chopped carrots, celery, and diced onion to the simmering broth. Cook for about 10 minutes until the vegetables become tender but still hold their shape, and the broth takes on a slightly richer aroma.
Stir in the leftover chicken pieces, letting them warm through gently in the broth for about 5 minutes. The chicken should be hot and tender, blending into the soup.
Season the soup with salt and pepper, adjusting to taste. Squeeze in fresh lemon juice for brightness, and add lemon zest for a subtle zing that lifts the flavors.
Stir in chopped parsley and drizzle with olive oil, giving the soup a final gentle stir to combine all the fresh, vibrant flavors.
Let the soup rest off the heat for about 5 minutes, allowing the flavors to meld and develop a comforting aroma. Then, ladle into bowls and serve hot, garnished with extra parsley if desired.