Heat the oil in a large heavy-bottomed soup pot and sauté the breasts until they are light brown, (about 5 minutes). Remove the breasts and set aside. Add half of the onions and sauté until translucent, (approximately 4 to 5 minutes). Remove and set aside.
Cut up the remaining parts of the chicken (not the breasts) into small pieces to allow them to release their juices in the shortest time possible. This is the hardest part of the recipe. If you have a meat cleaver, it makes the job easier, but if you don’t, do the best you can with a chefs knife.
Add the pieces to the pot, and cook for 8 to 10 minutes until no longer pink. Return the onion to the pot, reduce the heat to low, cover, and simmer until the chicken releases its juices, about 20 minutes. Add boiling water, reserved chicken breasts, 2 teaspoons of salt, and bay leaves. Cover and simmer until chicken breasts are cooked, about 20 minutes. Increase the heat if necessary.
Remove chicken breasts and set aside. Strain and reserve broth. Skim fat from the broth, reserving 2 tablespoons to be used to cook the vegetables. Add the reserved fat to the soup pot and sauté the remaining onions along with the carrot and celery for about 5 minutes. When the chicken breasts are cool enough to handle, remove and discard the skin and bones. Shred the breast meat into bite sized pieces and add to the pot. Add thyme and reserved broth; simmer until the vegetables are tender.
Season with salt and pepper, add parsley and serve. You can also add noodles, orzo, or small pasta shells and cook until tender.