Place the chicken in a large pot and cover with 16 cups water—enough to ensure the chicken is completely immersed. Add the bay leaves and salt. Bring to a boil. Reduce the heat and simmer, uncovered, for 1 hour 30 minutes or until the chicken is completely cooked through. Periodically skim the top of the broth to remove foam.
Carefully remove the chicken from the pot and transfer it to a large bowl.
Turn the heat on the pot to medium-low. Rapidly simmer the broth for an additional 20–30 minutes. It should be reduced to about 12 cups. It’s okay if you have a little more or a little less. Turn off the heat.
Strain the broth through a fine-mesh sieve into a large bowl. Wipe out the pot and return it to the stove. Rinse out the sieve and set it aside for the elbows.
Prepare the chicken:
Pick the meat from the bones and dice it or shred it with your fingers. Discard the bones and skin or reserve them for stock.
Make the soup:
Heat 1 tablespoon extra virgin olive oil in the pot over medium heat. Add the onion, leek, and celery, and season lightly with salt and pepper. Sauté for 6–8 minutes.
Add the garlic and thyme and sauté for 1 minute until fragrant. Add the potatoes and toss to coat. Add another pinch of salt and pepper and cook for 1 minute.
Simmer the soup:
Pour in 12 cups of prepared broth and bring to a boil. Add the diced chicken, reduce the heat, and simmer, uncovered, for 30 minutes or until the potatoes are fork-tender.
Make the elbows:
Bring a medium pot of salted water to a boil. Once boiling, add the elbows and cook until 1 minute under al dente, about 6 minutes. Drain through the sieve and add straight to the soup.
Finish the soup:
Taste the soup and season to your preferences. Add half the minced parsley and all of the lemon juice. Simmer for 5 minutes, then turn off the heat.
Make the garnish:
Combine the remaining minced parsley with lemon zest and Parmesan cheese, stirring to combine. Add a big pinch of black pepper.
To serve:
Ladle the soup into bowls. Spoon the Parmesan mixture on top of each bowl. Any extra Parmesan cheese garnish can be stirred back into the pot of soup before refrigerating. Enjoy!
Notes
If you decide to use a rotisserie chicken, skip straight to the "make the soup" part of the recipe. Shred the entire chicken, which should yield 4–5 cups of meat. Use it with 12 cups of store-bought broth.