In a medium bowl, whisk together the eggs and heavy cream until well combined and slightly frothy. Season with salt and pepper.

Melt the butter in a non-stick skillet over medium heat. Once the butter is melted and slightly sizzling, pour in the egg mixture.

Allow the eggs to cook undisturbed for about 30 seconds, or until the edges begin to set.

Gently push the cooked edges towards the center of the skillet with a spatula, tilting the pan to allow the uncooked egg to flow underneath. Continue this process until the eggs are mostly set but still slightly moist.

Sprinkle the shredded cheddar cheese and chopped bacon or ham evenly over the top of the eggs.

Fold the omelet in half using your spatula. Cook for another 1-2 minutes, or until the cheese is melted and the omelet is cooked through.

Slide the omelet onto a plate. Garnish with chopped chives, if desired.