Position a rack in the center of the oven and preheat the oven to 400 degrees F.
In a large skillet, heat the olive oil over medium-high heat. Sprinkle the pork on all sides with 2 teaspoons salt and 1 teaspoon pepper. Cook the pork, turning occasionally, until browned on all sides, about 8 minutes. Transfer the pork to a 13-by-9-inch glass or ceramic baking dish.
Add the mushrooms, garlic, onions, 1/2 teaspoon salt and 1/4 teaspoon pepper to the skillet and cook, stirring frequently, until the onions are soft, about 8 minutes. Stir in the flour and cook for 1 minute. Increase the heat to high. Add the wine and scrape up the browned bits from the bottom of the pan with a wooden spoon. Bring the mixture to a boil and cook for 2 minutes.
Pour the wine mixture into the dish with the pork and bake until a meat thermometer inserted into the thickest part of the pork registers 165 degrees F, 25 to 30 minutes. Transfer the pork to a cutting board and let rest for 15 minutes.
Pour the pan juices into a medium saucepan. Stir in the parsley and keep warm over low heat. Season with salt and pepper.
Cut the pork into twelve 1/2-inch-thick slices. Arrange in a single layer in the baking dish and place a piece of cheese on each slice. Bake until the cheese is melted, 3 to 4 minutes. Spoon the wine sauce over the top of the pork and serve.