Chicken noodle soup is much better when made with homemade broth and homemade noodles.
I like to make chicken broth after having a roast chicken - Throw the leftover chicken, bones and all, in a big pot with water and salt, enough to cover all the chicken. Let it cook at a slow boil until the chicken is falling off the bones. Then strain the broth off the chicken, return to heat and let it reduce until it has the "strength" of chicken flavor that you like.
While it's boiling down, you have things you can be doing. First off, mix up the noodle dough. Here is a small recipe. It can easily be doubled.
1.5 cup unbleached flour
1.5 tsp olive oil
1 Tbsp water
1/2 tsp sea salt
Put flour and sea salt in a bowl, then add beaten eggs, water, and oil. Mix with a fork and let the dough sit for at least 15-20 minutes, covered.
Okay, once your noodle dough is resting, start peeling and slicing carrots, slice celery, and dice onions; mince a little parsley, too, if you like. When your broth is reduced to about the strength you like, add the chopped vegetables. While the veggies are cooking in the broth, bone some of the chicken, which should have cooled by now. If you prefer nice chunks of chicken instead of little pieces of leftovers, you can use a leftover cooked chicken breast or, maybe you can just use your roast chicken entirely for the soup, cutting off some of the best of the chicken before it goes to making broth. The boiled bits of chicken are good for chicken tacos.
Now, roll out your noodles on a lightly floured surface. Turn them over every so often, dusting lightly with flour. Once they are quite thin, cut them in strips with a pizza cutter or knife.
Once your veggies are just right (not too mushy), add the chicken. Then, carefully add the noodles, a few at a time. (You may not want to use all of them, depending on how much broth you made. Just play it by ear.) Reduce the heat to medium-low and cover. Let the noodles cook for about ten minutes. Remove from heat and let cool a bit.