Peel the potatoes, cut them into small pieces and put them together with the vegetables in a pot, fill it up with water and cover it, add the broth, boil until the potatoes are done.
Skim the stock, but do not dump it, mash the potatoes, add milk and butter.
Fill up with broth according to taste (thin or thick soup), cut the sausages into small pieces and add them to the soup, let them infuse briefly.