In a bowl, whisk together the eggs, milk, salt, and pepper until smooth.
Heat butter in a nonstick skillet over medium heat until melted. Pour in the egg mixture, allowing it to spread evenly. Cook without stirring until the edges begin to set.
Sprinkle crabmeat, 1 tablespoon green onions, and cheddar cheese over half of the omelette. Cook until the cheese melts and the egg is set.
Fold the omelette in half over the filling. Slide onto a plate and serve hot with remaining green onions on top.