Crack the eggs into a mixing bowl and season with a pinch of salt and freshly ground black pepper. Whisk them vigorously with a fork until the whites and yolks are fully combined and no streaks remain, ensuring a uniform yellow color.

Heat an 8-inch non-stick skillet over medium-low heat and add the unsalted butter. Allow the butter to melt and foam, swirling it to coat the pan, but be careful not to let it brown as a French omelette should remain pale.

Pour the beaten eggs into the center of the pan and immediately begin stirring rapidly with a heat-proof spatula or fork while shaking the pan back and forth. This constant agitation creates small curds for a creamy, custard-like texture.

Once the eggs are mostly set but still look slightly runny and moist on top, stop stirring and use the spatula to smooth the surface. Run the spatula gently around the edges to ensure nothing is sticking to the pan.

Tilt the pan slightly and carefully roll the omelette down upon itself to form a neat cylinder or fold it in half. Slide the finished omelette onto a warmed plate, seam side down, and sprinkle generously with fresh chives before serving immediately.