Preheat the oven to 350ºF. Lightly oil a 13 X 9-inch baking dish.
In a large sauté pan, heat the olive oil over medium heat. Add the onions and season with salt and pepper. Add the prosciutto and cook, stirring, until the onions are soft and lightly golden, about 5 minutes. Stir in 1/4 cup of the parsley, the shallots, garlic, and mushrooms. Season with salt and pepper and cook, stirring, until the mushrooms are slightly limp, about 2 minutes. Add the wine and cook, stirring, for 6 minutes. Add the tomatoes, season with salt and pepper, and cook, stirring, for 10 minutes, then season again with salt and pepper and add the basil and oregano. Stir to mix. Remove from the heat.
In a large mixing bowl, combine the ricotta cheese, egg, remaining 1/4 cup parsley, 1/2 cup of the grated Parmigiano, grated mozzarella, and heavy cream. Season with salt and pepper.
Spoon 1 cup of the mushroom-tomato mixture into the baking dish and spread evenly over the bottom. Then arrange the Idaho potatoes in a single layer, slightly overlapping. Spread 1 cup of the cheese mixture over the potatoes. Repeat with another cup of the mushroom-tomato mixture, then a layer of sweet potatoes, 1 cup of the cheese mixture, then another cup of mushroom-tomato mixture, a layer of red potatoes, and another layer of the cheese mixture; finish with a layer of the mushroom-tomato mixture. Sprinkle with the remaining 1 1/2 cups Parmigiano. Cover the lasagna with a sheet of aluminum foil and bake for 30 minutes. Remove the foil and bake until the top is golden brown, 10 to 15 minutes.
Remove from the oven and let rest for 10 minutes before cutting. Place a portion of the lasagna in the center of each plate. Garnish with grated cheese.
Yield: about 8 servings