Brush a baking sheet lightly with oil and place under a preheated very hot grill to heat up.
Put the trout fillets on to the hot sheet, sprinkle with salt and pepper and place under the grill for 8–10 minutes until lightly browned and the fish flakes easily when pressed with a knife.
Meanwhile, put the basil and garlic into a bowl. Work in the oil using a hand-held blender. Stir in the Parmesan cheese.
Remove the fish from the grill, transfer to serving plates, drizzle with the pesto, sprinkle with extra basil leaves to garnish and serve with salad.