Preheat the oven to 375°F.
At least 1 hour before you plan to cook the fish, prepare the stuffing. Combine the shrimp, sage, softened butter, lemon juice, ½ teaspoon salt, ½ teaspoon pepper, paprika, scallions, breadcrumbs, and pine nuts in the bowl of a food processor fitted with the steel blade and pulse to combine well. Transfer the stuffing to a bowl, cover, and refrigerate until ready to use.
Season the trout with salt and pepper and rub the skin with the oil.
Divide the stuffing among the 4 fish, spreading an even layer inside the cavities. Close the fish to seal in the stuffing. Lay the fish in an oiled roasting pan. Roast in the oven until the fish is thoroughly cooked, 15 to 18 minutes. About halfway through the cooking time, carefully run a spatula under each trout to ensure it doesn’t stick to the pan.
Put 1 trout on each of 4 dinner plates. Drizzle any juices and melted butter from the pan over the fish and serve.