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Heat oven to 400F.
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Peel and dice the potatoes into ½ inch cubes.
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Heat the oil in a large fry pan over a medium heat.
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Add the onion and the sausage and sauté for about 3-4 minutes, until the onion is soft. Remove from the pan, but leave the oil in the pan.
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Add the potatoes to the fry pan and cook until soft – about 20 minutes. Keep stirring the potatoes to ensure they don’t burn.
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Return the sausage and onions to the pan and flatten the hash into an even layer, pressing gently with a spatula. Cook without disturbing if for 5 minutes or until the bottom is crispy. Flip the hash in sections and cook another 5 minutes.
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Make small indentations in the hash for the eggs. Crack an egg in to each indentation - transfer to the oven and bake until the egg whites set, about 10 minutes.
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Sprinkle grated cheese over the top and cook for another couple of minutes until melted.
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Season with salt and pepper and serve immediately.