Rinse the half courgette and grate it coarsely over a sieve or clean kitchen towel. Squeeze out the excess moisture from the courgette firmly to prevent the omelet from becoming too wet. Break the six organic eggs into a bowl, add the pinch of turmeric, and beat everything together well with a fork or whisk.
Heat a frying pan on medium-high heat and melt the tablespoon of ghee. Add the grated and squeezed courgette and sauté it briefly in the fat for one to two minutes. Then pour the beaten egg mixture evenly over the courgette in the pan. Reduce the heat slightly and let the omelet cook gently until the top is almost completely set and the bottom is cooked through.
Remove the pan from the heat once the eggs have reached the proper doneness. Crumble the fifty grams of feta with your hands and distribute the cheese over the top of the warm omelet. Season the dish to taste by sprinkling Celtic sea salt evenly over it.
Carefully slide the omelet from the frying pan onto a large plate. Fold the omelet in half if desired with a spatula and serve the dish immediately while it is still warm.