Mix the rösti with the cream cheese and Gruyère, shape into galettes (approx. 8 cm in diameter). Heat the clarified butter in a non-stick frying pan, then reduce the heat. Fry the galettes in batches over a medium heat for approx. 4 mins. on each side. Remove, keep warm.
Combine the vinegar and oil, season. Add the herbs, mix and spread on top of the rösti nests.
Heat a little clarified butter in the same pan. Crack the eggs one at a time and slide into the pan, fry over a low heat for approx. 3 mins., season with salt and serve with the nests.