In a medium bowl or the jar of a blender, whisk together or blend eggs, salt, pepper, cayenne, and heavy cream until well blended and frothy.
In a small nonstick skillet, melt butter over medium-high heat. Add ham and cook until browned, 1 to 2 minutes. Add egg mixture and stir lightly until just set on bottom, about 30 seconds. Sprinkle green onions across omelet and continue to cook for 30 seconds to 1 minute, shaking pan to loosen the eggs around edges.
When eggs are nearly completely set but still slightly soft, sprinkle cheese over top. Hold the pan in one hand and, with a flick of the wrist, flip omelet, then fold one-third toward center. Turn omelet out onto a plate, folding it over onto itself into thirds. Garnish with additional chopped green onions if desired and serve immediately.