Step 1

Prepare a grill for medium-high indirect heat (for a charcoal grill, bank coals on one side of grill; for a gas grill, leave one or two burners off). Grill eggplant over indirect heat, turning occasionally, until completely tender, 1–1½ hours (depending on size and type).

Step 2

Meanwhile, grill whole garlic heads over indirect heat until cloves are tender and jammy, 40–50 minutes.

Step 3

Mix together yogurt and tahini in a small bowl; season with salt. Slice heads of garlic in half crosswise and squeeze cloves into yogurt mixture; stir well to incorporate.

Step 4

Cut a slit lengthwise down each eggplant and carefully pull open with your hands. Arrange on a platter and squeeze juice from lemon over; season with salt. Dollop yogurt sauce around and sprinkle with sumac. Serve with spoon sauce (optional).