In a large pot, heat oil over medium-high heat until shimmering. Add chicken carcass pieces and cook, turning occasionally, until bones and any remaining skin are browned.
Add a splash of the water to deglaze the pan and scrape all the browned bits off the bottom. Add the vegetables, remaining water, and salt. Bring to barely a simmer (watch closely to make sure it never comes to a boil as that can make your soup cloudy).
Simmer for 90 minutes, scraping fat off the top occasionally, and checking to make sure your broth has not started to boil.
Discard large pieces of bone and vegetable before straining broth through a fine-mesh strainer. Rinse pot to remove any chicken or vegetable remnants and return broth to pot. Bring to a simmer and add carrots, celery, onion, and spices. Simmer for 20 to 25 minutes, until vegetables are tender.
While your vegetables cook, bring a pot of generously salted water to boil and cook noodles according to package instructions. Drain and set aside.
Stir cooked, drained pasta and cooked chicken into the broth and vegetables. Add more salt and pepper to taste. Bring to a simmer and cook for 1 minute to make sure the chicken is heated through.