Beat the eggs in a bowl. Mix in the milk or water, salt, pepper and green onion, leaving little onion behind to sprinkle on the omelette once cooked.
Melt the butter in an 8 or 9 inch non-stick skillet set over medium heat. Pour in the eggs. As mixture sets at the edges, use a heatproof spatula to gently push the cooked portions towards the centre of the pan. When the surface of the eggs is almost set but still looks moist, top one half of the omelette with the shrimp and cheese, leaving a bit of both to decorate the top of the omelette. Fold the omelette in half and cook 1 minute more. Slide onto a plate, decorate top with remaining green onion, shrimp and goat cheese, and serve.