Melt 1 1/2 tablespoons of the butter in a large heavy skillet over medium heat. Add the carrot, leek, and celery. Season with salt and pepper. Cook gently, stirring often, until softened, 5 to 7 minutes. Remove from heat and cool.
Preheat the toaster oven to 425 degrees F. Cut out a 13 by 15-inch parchment paper rectangle. Fold in half crosswise. Unfold and brush the whole rectangle lightly with some oil. Lay the fillets on one half of the parchment, skinned side down, and season with salt and pepper. Drain the vegetables of as much liquid as possible and divide among the fillets. Drizzle with the lemon juice. Top each vegetable mound with 1 teaspoon butter and drizzle each with 1 teaspoon of the truffle oil.
Fold the other half of the paper over the fillets and, starting with the top, make 1/4-inch folds all around the perimeters and press to crimp and seal. Close the final crimp with a paper clip. Place the packet on a baking sheet and bake until puffed and lightly browned, 10 to 12 minutes. Transfer to a warmed serving plate and serve at once, preferably while the packet is still puffed.
Recipe courtesy of Sara Moulton, Sara Moulton Cooks at Home, Broadway Books, 2002