Add coconut oil to a non-stick pan and heat over medium-high heat.
Meanwhile, add eggs, orange zest, salt, pepper and coconut milk to a small mixing bowl and give this a good whisk, until slightly frothy.
Pour that mixture into the preheated pan and swirl it around to spread your omelette evenly over the entire surface of the pan. Let the omelette settle for a few seconds then run a heat resistant rubber spatula around the edge to sort of break it down a little bit, then start swirling again. You want some of the uncooked eggs from the top to get under the omelette. Don’t be afraid to break it up a little bit. Repeat that process 2 or 3 times until the top really starts to set. Remove from heat and let your omelette sit for a few minutes until the top is completely (or almost completely) set. You can cover it if you want to speed things up a bit...
Delicately slide your omelette onto serving plate, (fold it in half while doing so if desired) and garnish with fresh (or dry, or toasted) unsweetened coconut shavings.