Pat chicken pieces completely dry with paper towels, then season all sides generously with 1 tsp salt and black pepper. Dice the onion, slice the carrots and celery into ¼-inch pieces, and mince the garlic.
Heat olive oil in a large Dutch oven over medium-high heat until shimmering. Working in batches, place chicken skin-side down and cook undisturbed for 5-7 minutes until deeply golden, then flip and brown the other side for 5 more minutes. Transfer to a plate and set aside.
Reduce heat to medium and add the onion, cooking for 5 minutes until translucent. Add carrots and celery and cook 4-5 minutes more, then stir in the garlic and cook for 30 seconds until fragrant.
Pour in chicken stock and water, scraping up all the browned bits from the bottom of the pot. Return the browned chicken (and any juices) to the pot, then add the bay leaves, thyme sprigs, peppercorns, and remaining 1 tsp salt.
Bring to a boil, then immediately reduce to a gentle simmer and partially cover with a lid. Skim any foam that rises to the surface during the first 20 minutes, then simmer for 1 to 1½ hours until the chicken is fall-off-the-bone tender.
Use tongs to remove all chicken pieces to a large bowl and let cool for 10 minutes. Discard bones, skin, and cartilage, then shred the meat into bite-sized pieces using two forks. Also discard the bay leaves and thyme sprigs from the soup.
If adding noodles or rice, bring the soup back to a boil and cook them according to package directions (8-10 minutes for egg noodles). Return the shredded chicken to the pot, taste and adjust salt and pepper, and heat through for 2-3 minutes.
Ladle the hot soup into bowls and garnish with freshly chopped parsley. Add a squeeze of fresh lemon juice to individual bowls for brightness if desired.