Place a pot over high heat. Add 8-10 cups water, chicken breasts, 4 whole garlic cloves, 1/3 of the carrots, all of the celery, the large onion pieces, bay leaf, celery leaves, whole parsley, whole thyme, and the pepper.
Bring just to a boil and reduce heat to medium-low. Allow to simmer for 30 minutes.
Using a colander set over a pot or large bowl, dump in broth so the colander catches all of the large ingredients and save the broth. Remove chicken and allow to cool on a plate so it will be cool enough for shredding. Discard skin.
Shred chicken with two forks or hands when cool enough to handle. Meanwhile, rinse the empty stockpot. Add olive oil over medium heat. Add chopped onions and carrots. Sauté for 5 to 8 minutes, until the onion has softened and is translucent. Add garlic and cook for an additional 30 seconds. Add stock, potatoes, salt, and chopped thyme. Allow to cook over medium heat for 10 minutes.
Add peas and shredded chicken. Add lemon juice to serve, if desired, and sprinkle with parsley.