Cook the potatoes in their skins until just tender, remove the skins while still warm and slice.
Sauté the onion and garlic in olive oil.
Add the chopped tomatoes and tomato puree and cook on a low heat for about 20 minutes.
Season to taste with salt and freshly ground pepper.
Grease a baking dish with butter. Pour half of the tomato sauce and sprinkle with a little grated cheese. Arrange the potatoes in a brick-like layer and season with salt and pepper. Spread the remaining tomato sauce evenly between and over the potatoes. Sprinkle with the remaining Sbrinz.
Cut the mozarella into cubes, the salami into strips and spread evenly with the olives on the gratin.
Bake the gratin on the lowest shelf of the oven heated to 220 degrees for about 20 minutes.