Separate the eggs, placing the whites into a large clean bowl and the yolks into a separate smaller bowl. Add a pinch of salt to the whites.
Beat the egg whites with a hand mixer on high speed until stiff, glossy peaks form, about 2 to 3 minutes. They should hold their shape firmly when you lift the beaters.
Add the softened cream cheese to the egg yolks along with a pinch of pepper. Whisk vigorously until the mixture is completely smooth and creamy with no lumps remaining.
Pour the yolk mixture over the whipped whites. Using a spatula, gently fold together with slow sweeping motions from the bottom up. Fold just until combined, being very careful not to deflate the whites. The batter should be airy and cloud-like.
Melt butter in a non-stick skillet over medium-low heat, swirling to coat the pan. Pour in the fluffy egg mixture and gently spread it into an even layer with the spatula. Cook undisturbed for 3 to 4 minutes until the bottom is golden.
Sprinkle the shredded gruyere over one half of the omelette. Cover the skillet with a lid and continue cooking for another 2 to 3 minutes until the cheese is melted and the top of the omelette is just set but still pillowy and jiggly.
Carefully fold the omelette in half using the spatula, then slide it onto a plate. Top with fresh chives and serve immediately while it is still tall and puffy.
