Instructions
Place the chicken, whole carrot, whole onion, garlic clove, and bay leaf in a large pot. Pour in enough water to cover by 3 inches and bring to a boil over medium-high heat. Season lightly with salt and pepper. Turn the heat down to a gentle simmer, cover partially, and cook for about 1½ hours, or until the chicken is falling off the bones.
Remove the chicken pieces and vegetables with a slotted spoon and set aside. Discard the vegetables and bay leaves. Let the chicken cool until you can handle it, then shred or pull the meat into thin threads; discard the skin and bones. Strain the broth through a fine-mesh sieve or cheesecloth-lined colander into a bowl.
Wipe the pot dry, but don't wash it. Add 2 tablespoons of the olive oil and heat over medium heat. Add the chopped carrots, minced onion, potato, and celery, cover, and cook, stirring on occasion, until softened and translucent, about 8 minutes. Return the shredded chicken to the pot and pour the broth back in. Simmer for 45 minutes, or until the vegetables are very soft.
Whisk together the remaining 4 tablespoons olive oil and the lemon juice and stir that into the soup. Season to taste with additional salt and pepper, stir in the parsley, and serve.