Prepare the Ingredients:

• Gather all ingredients and dice the ham, onion, and green bell pepper.
• Beat the eggs in a small bowl until just combined. Avoid overbeating, as this can affect texture.

2. Cook the Filling:

• Melt butter in a skillet over medium-high heat.
• Add the diced ham, onion, and bell pepper.
• Season with salt and pepper, and sauté for about 5 minutes, or until the onions soften and the ham begins to caramelize.

3. Cook the Omelet:

• Reduce the heat to medium-low and pour the beaten eggs into the pan.
• Mix briefly with a spatula while shaking the pan to evenly distribute the ingredients.
• Run the spatula along the edges to ensure even cooking.

4. Add the Finishing Touches:

• Sprinkle shredded Cheddar cheese and cayenne pepper over the omelet.
• Continue cooking until the top is still slightly wet but not runny, about 5 minutes.
• Fold the omelet in half with a spatula and transfer it to a plate.