Place the chicken, onion, carrots, celery and water into a large stock pot. Bring to a boil and then reduce to a simmer, uncovered. Let simmer for 2 1/2-3 hours. Remove the chicken, onion, carrot and celery. Increase the heat and bring the stock to a boil.
Add the salt and linguine to the stock. cook until tender, about 15 minutes. Decrease heat to low.
Meanwhile, cut the onion, carrots and celery into bite sized pieces (or the preferred sized). Shred the chicken, and reserve 2 to 2 1/2 cups of meat.
Once the noodles are cooked, add the butter, chicken and vegetables into the pan. Stir and let warm through for five minutes. Ladle into warm bowls and serve warm.
If you would like to speed up the chicken stock process, use this Instant Pot Chicken Stock recipe. To make this gluten-free, use your favorite gluten-free noodles.