Preheat a gas grill to medium-high (400°F to 450°F) on 1 side; keep the other side unlit. Remove any damaged outer leaves from cabbage. Keeping core intact, cut the cabbage into 6 wedges. Brush the wedges with oil; sprinkle with ½ teaspoon salt. Place cut-sides down on oiled grates over the lit side of the grill. Cover and grill, turning once, until both cut sides are well charred, 6 to 8 minutes per side. Flip onto rounded side; grill, covered and undisturbed, until well charred, about 3 minutes. Transfer to the unlit side; drizzle the wedges with 2 tablespoons water. Increase the heat to very high (500°F to 550°F); cover and grill until tender-crisp, about 25 minutes. Transfer to a plate and cover tightly with foil; let rest for 5 minutes.
Meanwhile, stir butter, cilantro, ginger, lime zest, garlic, crushed red pepper and the remaining ¼ teaspoon salt together in a small bowl. Transfer the cabbage to a platter; spread with the butter mixture.
