Coarsely grate the potatoes and pat dry between several sheets of kitchen paper. Mix thoroughly in a bowl with the onion, oregano and plenty of seasoning.
Melt the butter with the oil in a medium-sized, heavy-based frying pan. Tip in the rösti mixture and spread it out in an even layer, pressing down gently to compact it.
Cook over a very gentle heat for about 10 minutes or until the underside has turned golden. Test by lifting at the edge. To turn the rösti, invert it on to a plate and then slide it back into the pan to cook the base for a further 5–10 minutes until crisp and golden.
Arrange the tomato slices on top, seasoning with a little pepper. Sprinkle the Gruyère over the tomatoes and arrange the mozzarella slices on top. Sprinkle with the Parmesan and scatter with the olives.
Cook under a preheated moderate grill for about 5 minutes until the cheese bubbles and begins to colour. Garnish with basil leaves and serve with a green salad.