In a large skillet over high heat, cook the sausage and onion, stirring often, until the meat is browned and the onion is lightly golden, about 6 minutes. Remove from the pan and drain on paper towels. Set aside.

Combine the eggs, salt, black pepper, and water in a medium-size mixing bowl and whisk to mix.

In a medium-size nonstick skillet over medium heat, melt 1 tablespoon of the butter. Pour half of the egg mixture into the skillet and stir it around with a spatula, then swirl it around until it begins to set, about 3 minutes. Add half of the sausage-and-onion mixture, half the cheese, and half of the green onions, spreading them evenly over the eggs. Cook for 2 minutes. With a spatula, carefully fold the omelette in half and cook for 1 minute more. Carefully slide the omelette onto a serving plate, flipping it over as you do; keep warm.

Repeat the process with the remaining 1 tablespoon butter and the remaining egg mixture, sausage and onion, cheese, and green onion.

To serve, cut each omelette in half.