Rinse the smoked trout and remove any skin or bones. (If your smoked trout isn't "ready to eat," you'll need to poach it before you do any of this.) Flake the trout and set aside.
In a small saucepan, heat 1 tablespoon of the butter and the 1/4 cup milk over medium heat. When the milk starts to steam, add the fish, stirring occasionally. When the mixture comes up to a boil, remove the saucepan from the heat, cover and set aside for 20 minutes.
Preheat broiler. Beat the egg whites to stiff peaks and set aside. Beat the heavy cream into the egg yolks and season with pepper. Add the fish/milk mixture and stir together. Fold in the egg whites.
In an omelette pan over medium heat, melt the remaining tablespoon of butter. When the butter is bubbly, pour in the omelette mixture and cook (without stirring or flipping) until underside is golden brown. Remove pan from the heat, sprinkle cheddar over the top of the omelette and place pan under the broiler. Broil until cheese is browned.