Cook peeled potatoes in salted water until tender. Drain them very thoroughly, mash until smooth, and allow to cool.
Add the egg yolk, salt, and flour to the potato mixture. Mix just until a soft dough forms. Use the dough immediately, as specified in the original recipe.
On a lightly floured surface, roll the potato dough into a thick rope and cut it into equal portions.
Flatten each piece into a small round.
Lightly brush the center with beaten egg white and place a spoonful of grated cheese in the middle.
Bring the edges together and seal firmly. Shape into either round or small elongated dumplings.
Bring a large pot of salted water to a gentle boil. Add the dumplings and reduce the heat so the water simmers rather than boils vigorously.
Once the dumplings rise to the surface, cover the pot and cook for another 5 minutes.
Remove with a slotted spoon and drain well.
Melt the butter in a frying pan. Add the breadcrumbs and cook until golden and fragrant.
Arrange the potato dumplings on a serving dish and spoon the buttered toasted breadcrumbs over the top. Serve immediately.
