Heat the butter in a 4-quart or larger cooker. Stir in the onion, carrot, and celery, and cook over medium-high heat until the onion starts to soften, about 2 minutes. Add the broth, water, bay leaves, and chicken.
Lock the lid in place. Over high heat bring to high pressure. Reduce the heat just enough to maintain high pressure. Cook for 7 minutes (breasts) or 8 minutes (thighs, or a combination of breasts and thighs). Turn off the heat. Reduce pressure naturally for 4 minutes, then quick-release any remaining pressure. Remove the lid, tilting it away from you to allow steam to escape.
Set a colander over a large bowl, preferably one with a pouring spout (a 2-quart Pyrex measuring cup is perfect), and strain the soup.
Return the broth to the cooker. When the chicken is cool enough to handle, chop or shred the meat. If you wish, reserve the strained vegetables for the soup.
Bring the broth to a boil. Add the squash, cover, and cook until defrosted. (You can hasten process by breaking up block of squash with a fork.) Stir well to thicken soup. Lower the heat.
Stir in the chopped chicken, beans, corn, scallion greens, and tarragon or oregano with the cooked vegetables (if using), parsley, and salt to taste. Simmer until the noodles are done and the chicken is hot.
Stir in lemon juice, if desired, for a bright finish.