Grate the potato into a clean tea towel over a bowl and squeeze out the excess liquid. Season the grated potato with salt and freshly ground black pepper.
Heat the oil in a small frying pan. Place the grated potato into the pan, press down and fry for 3-4 minutes. Push half of the butter down the edges of the rösti to crisp the edges.
Carefully turn the rösti over using a fish slice and cook for a further 2-3 minutes, or until beginning to turn golden-brown. Push the remaining butter down the edges of the rösti and cook for a further minute.
To serve, place the rösti onto a plate, top with two fried eggs and sprinkle over the chives.