In a large bowl, whisk the eggs with the crumbled goat cheese, Parmigiano, parsley, chives, 1/2 cup of the Gruyère, 1 teaspoon of salt and 1/4 teaspoon of pepper. Pour the egg mixture into the skillet and cook over moderately low heat, stirring gently, until the eggs start to set, about 5 minutes. Sprinkle the frittata with the remaining 1/2 cup of Gruyère. Transfer the skillet to the oven and bake the frittata for 8 to 10 minutes, until the center is just set. Run a rubber spatula around the edge of the frittata and slide it onto a work surface. Let cool slightly, then cut it into 6 pieces.