Preheat oven to 350°F (175°C).
Flake smoked trout into bite-size pieces with a fork.
In a large bowl, whisk together eggs, heavy cream, green onion, chives, tarragon, smoked trout, and Parmesan. Season with a pinch of salt and pepper. Stir to combine.
In a 9- or 10-inch oven-safe skillet, melt butter over medium-low heat. Once melted, add egg mixture and cook, stirring constantly, until egg begins to set, 8 to 10 minutes.
Transfer skillet to the oven and bake until center is firm, about 20 minutes.
Cut into wedges and serve topped with sour cream and parsley.