Begin making the sauce by placing a heatproof bowl in a water bath in which the water is barely simmering. Separate two of the eggs and place the yolks into the bowl and whisk briskly for 3 minutes or until the egg yolks are fluffy and twice the size. Add the coconut oil in batches of two, one teaspoon at a time and continue to whisk between each addition. Whisk the sauce for about 7-8 minutes or until the mixture resembles a moderately whipped or thickened cream consistency.
Make your omelette by placing the remaining egg and egg whites into a bowl and whisking with salt, pepper and chives and pouring into a non-stick pan. Flip once before laying on a plate. Work quickly, keeping an eye on the sauce and continuing to whisk it while you’re working on the omelette. Keep it mildly warm but not hot, you do not want the eggs in the sauce to cook or harden.
Finish the sauce by adding the lime juice at the last minute and continuing to whisk briskly. Pour over your omelette roll immediately and serve with chives and more seasoning to taste.