Peel potatoes and cut into small pieces. Slice onion thinly.
In a heavy pot, melt butter at medium heat and cook onion until tender. Sprinkle flour and cook for a minute. Add chicken broth and potatoes, cover, and cook for 10-15 minutes until the potatoes get soft.
Transfer the soup to a blender, and process until very smooth. Put the soup back in the pot, and add milk, salt, and pepper. Stir and cook until just before boiling.
Serve with chopped parsley.