Grate the potatoes using a box grater and place on a clean tea towel. Wrapping them in the towel, squeeze out all the excess moisture, transfer to a bowl and season to taste with salt and pepper.
Melt the butter in a large nonstick frying pan. Divide the potato mixture into quarters and form each into a 10 cm (4 inch) cake. Add to the pan and cook over a medium heat for 5–6 minutes on each side until lightly golden.
Meanwhile, poach or fry the eggs.
Serve 2 röstis per person, topped with an egg and with a slice of smoked ham and some tomato ketchup.