Build a medium fire in a charcoal grill, or heat a gas grill to medium-high. (If using charcoal, mound ash-covered coals on one side of grill and rake a shallow layer over bottom of grill to maintain medium heat. Start a batch of coals burning in a charcoal chimney to replenish coals when needed.) Thoroughly clean grill grate; brush with oil.
Season cut side of fish with salt and pepper; brush with butter. Grill fish, cut side down, over shallow layer of coals (if using charcoal) until flesh is cooked halfway through, 10–18 minutes depending on the size and thickness of the fish. (Use long tongs to rake more coals underneath fish and maintain medium heat.).
Using two metal spatulas, gently turn fish. Cook, scales side down, until fish is cooked through, 10–18 minutes longer, depending on size and thickness of fish. (The flesh will begin to separate from the skin. Insert a small knife into flesh; if it slides through without resistance, fish is done.).
Transfer to a platter and serve.