How to Make It
In a large saucepan, cover the potatoes with water, bring to a boil and cook until tender, about 20 minutes; drain. Return them to the pan and shake over high heat for about 1 minute to dry. Pass the potatoes through a ricer into a large bowl. Stir in 1 tablespoon of the olive oil and season with salt and pepper.
Preheat the oven to 400°. Generously butter a 9-inch springform pan. Spread half the potatoes in the pan and top with the cheese and prosciutto, leaving a 1/2-inch border of potatoes all around. Cover the filling completely with the remaining potatoes. Smooth the top and brush with the remaining 1/2 tablespoon of olive oil. Bake for 25 minutes. Transfer to a rack to cool to room temperature.
In a large cast-iron skillet, heat a thin film of olive oil. Unmold the potato cake and carefully transfer it to the skillet. Cook over low heat until browned on the bottom, about 15 minutes. Invert the cake onto a flat plate, add another thin film of oil to the skillet and slide the cake back in. Cook over moderately low heat until browned on the bottom, about 10 minutes. Slide the cake onto a large serving plate, cut into wedges and serve.