Heat county ham in a small skillet over medium heat just until browned, 1 to 2 minutes per side. Remove from the heat, dice into small pieces, and set aside.

In a medium bowl, whisk together the eggs, salt, pepper, cayenne, and heavy cream until well blended and frothy.

In an 8-inch, nonstick skillet, melt the butter over medium-high heat. Add the egg mixture and stir lightly until just set on the bottom, about 30 seconds. Sprinkle the ham and green onions across the omelet, and continue to cook for an additional 30 seconds to 1 minute, shaking the pan to loosen the eggs around the edges.

When the eggs are nearly completely set but still slightly soft, sprinkle the cheese over the top of the omelet. Hold the pan in 1 hand, and with a flick of the wrist, flip, and fold 1 1/3 of the omelet toward the center. Turn out onto a plate, folding the omelet onto itself into thirds. Garnish with additional chopped green onions if desired and serve immediately.