Place chicken pieces in the bottom of the stock pot, thighs on the bottom, then the legs, then the breasts. Sprinkle with salt and pepper and cover with water. Bring to a boil then reduce to low and simmer until the skin is pulled back and the meat is white, about 30 to 45 minutes.
Remove the chicken pieces and place on a cutting board. Skim the foam off the top of the soup. Add the celery, carrots, onion, and poultry seasoning. Cover and simmer until the vegetables are tender, about 20 minutes.
Allow the chicken to cool so that you don’t get burned while handling it. Remove the skin and cut away all the meat. Cut meat into small bite-sized pieces. Add meat back to the soup. Taste the broth about twenty minutes after the chicken has been added. Add salt, pepper, and poultry seasoning to taste. (Note: My boys like the dark meat, so I've started putting just the white meat back into the pot and saving the thighs and legs for them to nosh on.)
About fifteen to twenty minutes before you are ready to serve, cook the gluten-free linguine and pour the soup over it in the bowls. If you are using regular linguine, you can add the noodles to the soup and cook until just past al dente.