Preheat the oven to 300 degrees F. Use a 8 by 8-inch baking dish or one of similar size, and butter the bottom and sides with 1 tablespoon butter. Wash potatoes and thinly slice them, using a mandolin, to approximately 1/16-inch thick. Layer 1/5 of the sliced potatoes into the baking dish, making 2 thin layers of potatoes overlapping. Sprinkle with 2 teaspoons of the Parmesan cheese. Top with 1/2-cup fruit, salt, pepper and 1/2 cup cream. Repeat this step 4 more times, ending with the potatoes. Top the potatoes with the remaining cream and the 3 tablespoons of the butter cut into small pieces. Sprinkle with the rest of the cheese.
Cover the pan with foil and bake for 2 hours. Uncover the pan and bake for an additional 15 minutes until lightly browned the top. For a darker brown place the potatoes under the broiler for a few minutes. Remove the potatoes from the oven and let them sit for 15 minutes before serving. Cut into squares with a sharp knife.