Melt the butter in a large, well-seasoned cast iron skillet over medium heat. Add the potatoes and cook, stirring occasionally until tender and brown, about 15 minutes. Stir in the parsley, garlic, salt, and pepper, and remove from the heat.
Push the potatoes aside to make 4 evenly spaced shallow nests and break 2 eggs into each. Bake until the egg whites are cooked and the yolks are still runny, about 10 minutes. Sprinkle the cheese over the eggs and continue baking until it just melts, about 1 minute more. Serve immediately.