Melt the butter in a frying pan.
When hot, add potato and cook until golden-brown and crusty on all sides.
Beat up eggs in a small bowl with very little salt (the cheese has plenty) and freshly ground pepper.
Lower heat of stove, add eggs to pan and give them a good stir, a couple of times, bringing up cooked eggs from bottom, but redistributing uncooked eggs and potatoes.
Sprinkle generously with shredded cheese.
Let cook until golden-brown on bottom.
Run spatula down center, and carefully fold one half over the other creating a half-moon (sometimes you don't get a perfect half-moon, but it still tastes good anyway!).
Turn off heat and let omelette sit in pan for 30 seconds to a minute (depending on how well you want your eggs done) and then carefully turn over and allow 30 more seconds.
Allow to cool about 4-5 minutes before tucking in (this seems to allow the flavours to blend better).