To a medium saucepan, add the chicken breasts, chicken broth, thyme, oregano, rosemary, garlic, salt and a few grinds of black pepper. Bring to a boil then reduce the heat to low and cover with a tight-fitting lid. Gently simmer for 1 hour or until the chicken is fall-apart tender.
Meanwhile, make the soup base. Heat the olive oil in a large pot over medium heat. When shimmering, add the onion and sauté for 10-15 minutes, until caramelized. Add the celery and carrots then remove from the heat until the chicken is ready.
Transfer the chicken breasts to a plate and use two forks to shred into bite-sized pieces. Use a fine-mesh strainer to strain the cooking liquid into a large bowl. Save the broth, but discard the aromatics.
To the pot with the vegetables, add the shredded chicken, 3 cups of chicken broth and 1 cup of the cooking liquid. Add more cooking liquid by the 1/2 cup, tasting as you go, until the soup is salted to your liking.
Bring to a boil and cook for another few minutes, until the vegetables have softened. Stir in the parsley, dill and basil. Serve each portion with cooked pasta or rice and more freshly ground black pepper.