In a small bowl, beat together 3 of the eggs and season lightly with salt and pepper. Set aside.
Heat 3 tablespoons of the butter in a small non-stick skillet, with tight-fitting lid, over medium heat. Add the peppers and cook, covered, without stirring, until tender, about 3 minutes. Uncover and raise the heat to medium-high. Add the ham and spring onions and cook, stirring frequently, until hot, about 1 minute. Transfer the mixture to a bowl.
Melt a nut-sized piece of the remaining butter in the skillet over medium heat. When the foam subsides, add 1/4 of the pepper mixture. Pour in the beaten eggs and cook, stirring constantly with a heat-resistant rubber spatula, until just cooked, about 30 seconds. Turn off the heat and, using the rubber spatula, smooth over the top of the omelette so that it evenly covers the inside of the skillet. If using the cheese, sprinkle about 2 tablespoons over the omelette. Let rest for 30 seconds until the omelette has set.
Using the rubber spatula, fold the omelette in half, and transfer to a warmed plate. Cover with foil to keep warm. Repeat with the remaining ingredients to make 4 omelettes in all.
Cook's Note: If you like your eggs well-done, cover the pan with the lid after you've smoothed over the eggs, and let it rest for 30 seconds.