Grate the potatoes coarsely and squeeze any excessive moisture out of them.
Peel and finely slice the onion. Toss the potato and onion together in a bowl with salt and pepper and some thyme leaves.
Heat the oil in a large oven-proof pan and start to cook the potato mix over a medium heat for 10 minutes, turning occasionally.
Preheat an oven to 220°C.
Turn the potato mixture around in the pan one last time, then press down.
Place dots of butter on top and around the sides of the rösti. Bake the dish for 20 minutes.
Grate or chop the cheese and pull the ham apart into strips. Scatter both onto the cooked, golden and crispy rösti for a final 2 minutes of cooking.
Fry the eggs in a non-stick frying pan for 2 minutes so cooked but the yolk still runny. Transfer the eggs on top of the rösti dish, garnish with fresh thyme sprigs and a crack of black pepper then serve. Serves 2