Preheat the oven to 375 degrees F.
Heat a large skillet over medium heat and add the oil. When the oil begins to smoke lightly, add the butter and the potatoes. Season with a pinch of paprika and salt and pepper, to taste. Cook, stirring from time to time, until they become tender but crispy, about 10 minutes. Add the onions and the wine. Season again with salt and pepper and increase the heat. The potatoes should eventually start to brown. Add more butter, if desired. Cook for 5 to 8 minutes, stirring from time to time, so the potatoes brown on all sides.
Put the cheese in the middle of the skillet on top of everything and put the pan in the center of the oven. Bake until the cheese melts, about 8 to 10 minutes. The potatoes should be tender when pierced with the tip of a knife. Remove from the oven and season again with salt, if needed. Transfer to a serving platter and serve immediately.