Place the grated potatoes between some paper towels or a clean kitchen towel. Press down to release some of the moisture.
In a 10-inch heavy non-stick skillet over medium heat, warm 1 tablespoon of butter and 1 tablespoon of oil.
As soon as the butter has melted, press 1/2 of the potatoes into a single layer in the pan. Season with 1/2 of the salt and pepper.
Add 3 slices of the prosciutto to the potatoes. Sprinkle the grated cheese all over the prosciutto. Add the last 3 slices of prosciutto. Then, cover the filling completely with the remaining potatoes. Season again with salt and pepper and press down with a spatula.
Cook the potatoes, moving the pan around a couple of times, until the bottom is golden brown and crisp, approximately 9 minutes.
Loosen the potato cake gently with a spatula. Slide it onto a large plate. Place another plate on top of the potato cake and invert (turn upside down). Add the remaining butter and oil to the pan. As soon as the butter melts, slide the potato cake back into the pan, browned side up. Continue to cook until the underside is again golden brown and crispy and the cheese has melted, approximately 8 minutes longer.
Loosen the potato cake with a spatula and slide it onto a clean cutting board. Cut into 4 wedges and serve while hot.