In a non-stick skillet, heat 1 tablespoon butter and optionally 1/2 tablespoon olive oil over medium heat. Add diced ham, bell peppers, and onion, and sauté until the vegetables are tender and ham is lightly browned, about 3 to 4 minutes. Remove the mixture and set aside. Wipe the skillet clean with a paper towel.
Crack the eggs into a bowl, add a pinch of salt and pepper, and half of the chopped chives. Whisk vigorously until well combined and slightly foamy. Let the eggs rest for about 5 minutes to promote a light, fluffy texture.
Melt the remaining 1 tablespoon of butter in the skillet over low heat, tilting to evenly coat the bottom. Pour in the beaten eggs, spreading them evenly. As the eggs begin to set, gently lift the edges with a spatula and tilt the pan to let uncooked egg flow underneath.
When the eggs are mostly set but still slightly runny on top (about 4 to 5 minutes), spread the ham and vegetable mixture evenly over half of the omelette. Sprinkle shredded cheese over the filling.
Carefully fold the other half of the omelette over the filling to form a half-moon shape. Cover the skillet and cook on low heat for 1 to 2 minutes, allowing the cheese to melt thoroughly without overcooking the eggs.
Slide the omelette onto a plate and garnish with the remaining chopped chives, parsley, and if desired, sliced green onions. Serve immediately with your favorite sides such as hash browns, buttered toast, or fresh fruit.