Melt butter in a large cast-iron skillet over medium heat. Add potatoes and cook, stirring occasionally, until tender and brown, about 15 minutes. Stir in parsley and garlic. Season with salt and pepper, to taste, and remove from heat.
Make 4 shallow depressions in potatoes and break 2 eggs into each. Transfer to oven and bake until egg whites are cooked and yolks are still runny, about 10 minutes. Sprinkle cheese over eggs and continue baking until cheese melts, about 1 minute more. Serve immediately.
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