In a medium-sized bowl, crack the eggs and whisk them together with a pinch of salt and pepper. Set aside.
Heat 1 tablespoon of butter or oil in a non-stick skillet over medium heat. Add the diced ham, green bell pepper, and onion to the skillet. Cook, stirring occasionally, for 4-5 minutes, or until the vegetables are softened and the ham is slightly browned. Remove the mixture from the skillet and set it aside.
Wipe the skillet clean with a paper towel and add the remaining tablespoon of butter or oil, heating it over medium-low heat.
Pour the whisked eggs into the skillet. Allow the eggs to cook undisturbed for 1-2 minutes, or until the edges start to set. Use a spatula to gently lift the edges of the omelette, allowing the uncooked eggs to flow underneath. Continue this process until the omelette is almost set, but still a bit soft in the middle.
Spread the cooked ham, bell pepper, and onion mixture over one half of the omelette. Sprinkle the shredded cheddar cheese on top of the mixture.
Carefully fold the other half of the omelette over the filling, using a spatula to help guide it. Cook for an additional 1-2 minutes, or until the cheese is melted and the omelette is fully cooked through.
Gently slide the Western Omelette onto a plate and serve immediately. Enjoy!