Combine the vinegar, water, salt and sugar in a medium saucepan and bring to a boil. Place the nectarines and red onion into a jar or bowl. Pour the liquid over top, then let sit while you prepare the fish. Pickled nectarines can be made up to one week in advance. Store, covered, in the refrigerator.

Preheat a gas or charcoal grill to medium heat. Mix together the mustard, brown sugar, salt and pepper in a small bowl. Pat the barramundi fillets very dry, then rub the mixture all over on both sides. Lightly oil the tops of the cedar planks, place them on the grill, close, and allow them to heat up for about 5 minutes. Carefully transfer the fillets to the cedar planks (skin side down) close the grill, and turn the heat down slightly to medium-low.

Allow the fish to cook for about 10-15 minutes, depending on thickness. It will be done when the thickest part of the flesh is totally opaque throughout. Transfer the fish to a serving platter, then drain the pickled nectarines and red onions and spoon over the top. Drizzle with olive oil and sprinkle with chopped basil. Serve immediately.