Peel potato and grate with a fine box grater. Squeeze out as much of the juice as possible from the finely grated potato. Add a little salt and pepper.
Combine grated potato and shredded cheese. Divide into 4 equal portions.
Heat a large non-stick fry pan. Spread out each portion of grated potato and cheese mixture like a pancake of about 3½ inch diameter in the pan. Cook for 1 to 2 minutes. Gently flip over. This may be a little tricky as the mixture is very soft. Gently pat and reshape with spatula if they clump up. Cook for another 1 to 2 minutes or until lightly browned. Remove and place on a plate. Allow them to cool slightly.
Press each of the shredded cheese and potato pancakes into 4 indentations of an ungreased and unlined non-stick muffin pan to form a cup. Crack an egg into each cup. Add a little salt and pepper to each egg.
Bake for 15 to 18 minutes depending on how cooked you like your eggs to be.