Boil the potatoes in their skins in salted water until tender (about 25 minutes), peel while warm and mash until smooth with a potato masher.
While still warm, mix in the butter, the 2 egg yolks, the salt and the flour, and work into an even dough.
Whisk the 2 egg whites to stiff peaks and fold them gently into the potato mixture.
If using mushroom filling, fry the chopped mushrooms in 1 tbsp of butter in a frying pan, season with salt and cook for 5-6 minutes until all their moisture has gone, then leave to cool; skip this step for the ham filling.
With wet hands, shape the potato mixture into apricot-sized balls, press a little ham or mushroom into the middle of each, then close and shape into a dumpling again.
Bring plenty of salted water to the boil and carefully add the dumplings.
Cook for 10-15 minutes until they rise to the surface and are cooked through.
Meanwhile, toast the breadcrumbs in the rest of the butter in a frying pan until golden brown.
Lift out the dumplings with a slotted spoon, roll them in the toasted crumbs, then drizzle with sour cream to serve.
Based on: A jó házikonyha (a classic Hungarian cookbook), modernised. Calories and protein are estimated.