Put the chicken with the sliced vegetables in salted water, and cook, covered, until tender, about 35-40 minutes.
Remove the chicken and the vegetables from the broth; once the meat has cooled a little, take it off the bone, cut the meat into bite-sized pieces, and put it back in the broth together with the vegetables.
Make a light roux from the butter and the flour in another pot, pour in some of the chicken's cooking liquid, stir smooth, then add it to the chicken soup, and season with pepper.
Meanwhile make the dumpling dough: mix the egg with a pinch of salt and enough flour to make a thick batter, then pinch small dumplings off it with a spoon into boiling salted water, and cook for 2 minutes.
Bring the soup to a boil, and add the dumplings.
When serving, mix in the egg yolks, the sour cream and the butter, don't boil it any further.
Based on: Malatinszky Fanny szakácskönyve (a classic Hungarian cookbook), modernised. Calories and protein are estimated.