Preheat the oven to 400ºF. Preheat the grill.
In a medium-size mixing bowl, toss the carrot, parsnip, and sweet potato with 1 tablespoon of the olive oil, then season with salt and pepper. Put the vegetables on a baking sheet and roast until tender, about 40 minutes.
Put the Idaho potatoes in a large saucepan, cover with water, and season with salt. Bring to a boil, reduce the heat to medium and cook until fork tender, about 10 minutes. Drain and return to the saucepan and over low heat let the potatoes dry for 1 minute. Add the roasted vegetables and mash with a potato masher, leaving some small lumps. Season with salt and pepper. Add the heavy cream and 2 tablespoons of the butter and mix well. Season again with salt and pepper.
Drizzle the portobellos with 1 tablespoon of the olive oil and season with salt and pepper. Grill for 3 to 4 minutes on each side. Spread one quarter of the potato mixture over the gill side of 4 of the portobellos. Top with the remaining ones to form sandwiches.
Put the sandwiches on a baking sheet and bake for 6 minutes.
In a large sauté pan, heat the remaining 3 tablespoons olive oil over medium heat. Add the onions, season with salt and pepper, and cook, stirring, for 2 minutes. Add the sliced mushrooms and garlic, season with salt and pepper, and cook, stirring, for 3 to 4 minutes. Add the parsley and veal reduction and bring to a boil. Reduce the heat to medium-low and simmer for 2 minutes. Remove from the heat and stir in the remaining 2 tablespoons butter.
To serve, spoon some of the mushroom stew in the center of each plate. Place the Portobello sandwich on top of the stew, then top with more stew and garnish with the shaved cheese.
Yield: 4 servings