Make the stock as described in "Chicken Stock" recipe, others can be used as well, make sure it's salty. From the meat of a 5-pound chicken, I usually reserve the drumsticks for something else, but use the thighs.
Put the butter and onion into the bottom of a soup pot. Turn the heat to medium low and season with salt. Simmer, covered slowly for 10-20 minutes, until well wilted, but not brown.
Add the carrots. Coat well in the butter and onion. Cover and cook two to four minutes.
Add the celery. Coat well in the butter and onion juices. Season with some salt and some pepper lightly.
Cook, covered, stirring from time to time.
The goal is here to have the celery and carrots almost fully cooked without ever having introduced any broth to the pot. It gives them a good flavor, otherwise they get bland. The onions can dissolve into nothingness, it is for the taste, not the texture. Have the carrot done and the celery almost near completion.
Stir in the flour and cook briefly. Pour over the broth through a strainer, if it has not already been done.
Add the noodles. They will soak up a lot of salt, so even if the soup is salt, it may need a little bit more.
Cook the noodles only for a couple of minutes. They cook fast and should be al dente.
Add the chicken and some parsley. Stir well, tasting. Stir or sit until the noodles feel cooked right. Serve.
Note: If the vegetables feel hard a little after the broth is added, simmer them briefly before adding the paste. They will not cook much further during that time. Don't have them mushy. This soup may be a little hard to get the first time because there are a few processes going on, but it is good. The best noodle soup I know. Enjoy.