Peel and quarter the potatoes. Peel the onion and cut into fine strips.
Melt the butter in a large saucepan. Add the potatoes and onion strips and fry for 10 minutes with the lid closed (stir several times). Make sure that nothing burns! Now add the broth and salt and pepper. Bring to the boil with the lid closed and stirring constantly. Let the soup simmer for approx. 30 minutes until the potatoes are soft. Stir every now and then.
Take the saucepan off the stove and put everything through a sieve into a second large saucepan. Put the potatoes and some of the onions (depending on your taste, if you like onions you can save yourself the sifting) back to the broth and add the milk. Turn the stove back to full heat and puree the soup with a blender until it becomes a homogeneous, creamy mixture. Season again to taste with salt and pepper. Add the cornstarch and bring to the boil briefly. Serve with parsley and chives.
Onion bread tastes excellent with it. The soup can be prepared and reheated very well.