Are there any specific tricks or tips for cooking a classic French omelette in a carbon steel pan?
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The main issue in cooking a french omelet in carbon steel it'll stick too much. As I was taught you should just try to use a different pan but if you must use that kind then be sure to properly lubricate the pan with fat before adding egg. Then again if and when the eggs absorb it. Then to change it on to a lower heat than typically then making a french omlete so that it will not burn or get stuck as easily. Be sure to cook it long enough so that the bottom is firm enough not to break before moving it. With the excess fat it should release from the pan much like meat does. Once it has released from the pan be sure to keep it moving so that it will not stick again.