Wash the courgette under the tap and coarsely grate it with a box grater. Chop the fresh chives into fine pieces. Crack the six eggs into a bowl and beat them well with a fork. Mix the chopped chives through the eggs and season with pepper and salt. Squeeze out the excess moisture from the grated courgette with your hands.
Heat the olive oil in a large frying pan over medium-high heat. Add the grated courgette and cook for about two minutes until most of the remaining moisture has evaporated. Pour the egg mixture evenly over the courgette in the pan. Crumble the feta immediately and distribute the pieces over the still-wet eggs.
Lower the heat and place a lid on the pan. Let the omelette set gently for five to seven minutes. Check that the top is completely set. Remove the pan from the heat and let the omelette cool briefly. Slide the omelette carefully onto a flat plate and cut it into equal wedges to serve immediately.