Crack eggs into a mixing bowl, add salt, black pepper, and water; whisk until smooth and frothy. Fold in green onions, carrots, and ham.

Heat a nonstick skillet over medium-low heat, drizzle oil, and pour half of the egg mixture, swirling to cover. Cook for 2-3 minutes until edges set.

Fold one edge of omelet over itself when about 70% cooked, gently roll to the opposite edge while cooking for another minute.

Pour remaining egg mixture into the skillet, lifting the rolled omelet to let uncooked egg flow underneath. Cook for an additional 2-3 minutes.

Remove from heat and let omelet cool for 5 minutes to set flavors, then slice into 1-inch pieces.

Serve warm, optionally drizzling with soy sauce or sriracha for added flavor.