Crack the eggs into a bowl and whisk them together with a fork.
Add the diced onion, green pepper, and ham to the eggs and mix well.
Heat a non-stick skillet over medium heat and add the butter.
Once the butter has melted, pour the egg mixture into the skillet.
Use a spatula to gently lift the edges of the omelette and allow the uncooked egg to flow underneath.
When the omelette is almost set, sprinkle the shredded cheddar cheese over one half of the omelette.
Use the spatula to fold the other half of the omelette over the cheese.
Cook for an additional minute or until the cheese has melted.
Season with salt and pepper to taste.
Serve hot on the east side.
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