In a bowl combine the gingerroot, garlic, lime juice, oil, thyme, salt and pepper. Arrange the swordfish steaks in a shallow glass baking dish and spoon the marinade over them. Let the fish marinate, covered and chilled, turning once, for an hour. Meanwhile, make the pesto. In a food processor combine all the ingredients and transfer to a bowl.

Place swordfish steaks on your grill. Cook the steaks 4 inches above the coals for 3 to 4 minutes on each side, or until firm to the touch. Alternatively, broil them under a preheated broiler 4 inches from the heat for 3 to 4 minutes on each side. Serve with pesto.