In a bowl, crack two eggs and whisk them well with a fork, using right to left movements until they becomes homogenous. The purpose of the movement is just combining the yolks and the whites, not incorporating air in the mixture.
Warm up the pan over medium low heat and add the butter. Once it melts, move the pan in a circular motion so the butter can cover it all over.
When the butter bubbles, add your eggs to the pan and let cook over medium low heat. Mix with a spatula slightly, just until the eggs begin to set. Bring all the irregular edges in and let the eggs set. Instead of being cooked until browned, French omelettes are cooked just until set, resulting in a creamy, custard-like texture and a light exterior.
Add a pinch of salt on top of the eggs as well as your filling of choice and fold the omelette in half or thirds.
Sprinkle some extra salt, fresh ground black pepper, and chopped chives over the omelette and serve with mustard and a small side salad.