In a medium bowl, whisk together eggs, cream, and salt until fluffy and pale in color.
Heat a medium skillet over medium heat. Add bacon, and cook until crisp; remove bacon to paper towels to drain, reserving 2 teaspoons drippings in skillet. Crumble bacon.
Cook bell peppers, onion, and garlic in hot drippings over medium heat, stirring occasionally, for 10 minutes or until vegetables are tender. Add ham; cook for 1 minute or until heated through. Stir in crumbled bacon.
In an 8-inch nonstick skillet, melt 1/2 tablespoon butter over medium-low heat. Cook for 2 to 3 minutes or until butter starts to foam. Add half of egg mixture to pan; cook for 1 minute (do not stir).
Gently lift edges of omelette with a spatula, and tilt pan so uncooked portion flows underneath. Cook for 2 to 3 minutes or until center is almost set. Add half of vegetable mixture and 1/3 cup cheese to one side of omelette. Fold in half, and transfer to plate
Repeat with remaining butter, egg mixture, vegetable mixture, and cheese. Serve immediately.