Season the Chicken - Season both sides with salt and pepper.
Cook the Meat - Heat a large pot or dutch oven over medium heat. Add 1 tablespoon of olive oil. Once hot, place the chicken skin-side down. Cook until golden brown, about 6 to 8 minutes.Flip and cook until the bones are browned, about 5 minutes. Transfer to a plate. There should be about 1 tablespoon of drippings left in the pan.
Saute the Vegetables - Add 1 tablespoon of olive oil to the pan and heat over medium heat. Once hot, add the onions. Stir and cook until softened and lightly brown, about 5 minutes. Add celery, and cook for 3 minutes. Add carrots, and cook for 2 minutes.Add garlic, thyme, and rosemary to the pan, stir and cook for 30 seconds. Add salt and pepper, and stir to combine.
Simmer the Soup - Add the chicken back to the pan, skin-side up. Add bay leaves and chicken stock. Bring the liquid to a boil, cover, and reduce to a simmer over medium-low heat. Reduce heat to low if needed to keep the stock at a simmer. Cook the soup until the chicken reaches an internal temperature of 160ºF (71ºC), about 15 to 25 minutes. Discard bay leaves and remove the chicken from the pot.
Shred the Chicken - Once cool enough to handle, remove the skin and bones. Shred the chicken into small pieces using two forks.
Add the Noodles - Turn the heat to medium-high. Once the stock is at a rapid simmer, add the pasta. Stir and cook until tender, 5 to 7 minutes. Turn off the heat. Add the shredded chicken and allow it to warm for a few minutes. Taste and season with salt and pepper as needed.
To Serve - Garnish the soup with chopped parsley and serve hot.
Using Boneless Chicken Breast or Thighs: Sear the surface and check for doneness earlier as they cook quicker. The soup will not have as much body with boneless cuts.