Step 1Heat the oven to 400 degrees. Melt the butter in a large skillet over medium heat. Add the shallots and cook just until softened, 3 to 5 minutes. Add the mushrooms and cook, stirring occasionally, just until they have begun to soften and color, 3 to 5 minutes. Season to taste with salt.
Step 2Spoon half the mushrooms in a thin layer across the bottom of a 3-quart gratin dish. Season the fish lightly on both sides with salt and arrange it in a single overlapping layer on top of the mushrooms. Have the thinnest tail ends pointed to the ends of the dish and place the thickest pieces in the center. Spoon the remaining mushrooms over the top.
Step 3Moisten with the white wine (the fish and mushrooms will produce more liquid as they bake) and cover tightly with aluminum foil. Bake 15 minutes, then remove the foil, swirl in the whipping cream and return the dish to the oven until the fish flakes easily and the cream is bubbling, about 20 minutes more.
Step 4When the fish is cooked, sprinkle it with the minced parsley, chives and tarragon and serve immediately. Remove the fish bones before eating.