1. Place the fish in a saucepan and add enough cold water to cover. Bring to a simmer over medium heat, and simmer about 7 minutes, until the fish is hot and flakes easily. Drain. Remove the skin and bones and flake the fish.

2. Whisk the egg whites in a bowl until soft peaks form. Beat the yolks with 2 tbsp of the half-and-half in another bowl. Fold in the whites. Add the fish and 2 tbsp of the Parmesan. Season with pepper.

3. Preheat the broiler. Melt the butter in a large nonstick frying pan over medium heat. When foaming, add the egg mixture. Cook, lifting up the sides of the omelet as they set with a rubber spatula, allowing the uncooked egg mixture to run underneath the cooked portion, about 3 minutes, or until the omelet is almost set but the top is uncooked.

4. Sprinkle the remaining 2 tbsp of Parmesan over the omelet. Pour the remaining half-and-half over the top. Broil until the top of the omelet is set and lightly browned. Serve immediately, straight from the pan.