Put an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
Using the tines of a fork, prick the skins of the potatoes all over and arrange on the prepared baking sheet. Bake for 1 to 1 1/4 hours until tender and cooked through. Remove from the oven and let cool until warm enough to handle, about 20 minutes.
While the potatoes are cooling, in a medium skillet, heat 3 tablespoons of olive oil over medium-high heat. Add the onion and cook, stirring frequently, until soft, about 5 minutes. Add the mushrooms, garlic, 1/4 teaspoon salt and 1/4 teaspoon pepper. Cook until the mushrooms are golden and most of the liquid has evaporated, about 6 minutes. Put the mixture into a medium bowl. Cut the potatoes in half lengthwise and, using a small spoon, scoop out all but 1/4-inch of the flesh. Add the potato flesh to the mushroom mixture. Add the butter, oregano, and 1 cup of the cheese. Mix well. Season with salt and pepper, to taste.
Arrange the potato halves, cut side up on the baking sheet. Using a pastry brush, brush both sides of the potatoes with the remaining 3 tablespoons of olive oil and season the inside with salt and pepper, to taste. Spoon the mushroom mixture into each potato half. Arrange 1 half-piece of prosciutto on top and sprinkle with the remaining cheese. Bake until the cheese is melted and the prosciutto is crispy, about 20 to 25 minutes. Remove from the oven, transfer to a serving platter, garnish with chopped parsley and serve.