Place the large heavy-bottomed pot on the stove and add the olive oil. Turn the heat to medium and sauté chopped onions, carrots, and celery until fragrant and slightly softened, about 8 minutes, with the vegetables gently bubbling and releasing a sweet aroma.
Add the bone-in chicken pieces to the pot, then cover with cold water until the chicken is submerged. Increase the heat to high and bring to a gentle simmer, skimming off any foam that rises to keep the broth clear and clean smelling.
Drop in the herb sprigs—thyme and parsley stems—and a bay leaf if you have one. Reduce the heat to low and let everything simmer gently for about 1.5 hours, allowing the flavors to meld and the chicken to become tender and easy to shred.
Carefully remove the cooked chicken from the broth and set aside to cool slightly. Strain the broth through a fine mesh strainer into a clean pot, discarding solids and herbs, to ensure a clear, silky broth.
While the broth is hot, shred the chicken meat using two forks, removing bones and skin. Return the shredded chicken to the strained broth, stirring gently to combine.
Season the soup generously with salt and freshly ground black pepper, then squeeze the juice of a lemon into the broth for brightness. Taste and adjust the seasoning as needed.
Finish by stirring in chopped fresh parsley for herbal freshness, and serve the steaming soup in bowls. Enjoy the warm, comforting flavors with a slice of crusty bread if desired.
