Grilled Purple Cabbage Wedge Salad

Light a charcoal chimney starter filled with all-natural lump charcoal; let burn until completely gray, about 30 minutes. Dump charcoal into a fire pit or grill (under grill grate); let burn, uncovered, until smoldering, about 30 minutes. Fan the outer layer of ash off coals. Rub cabbage with 1 tablespoon oil; place directly on hot coals. Cover and cook, turning occasionally, until tender and charred on all sides, 2 hours and 30 minutes to 3 hours. Remove from coals; let stand until cool enough to handle, about 30 minutes. Discard outer charred leaves. Cut cabbage into 6 wedges; remove core. Whisk together yogurt, mayonnaise, buttermilk, lemon juice, garlic, salt, and remaining 1 tablespoon oil in a bowl until smooth. Drizzle over wedges. Sprinkle with feta, chives, and sumac.