Prepare a grill for medium-high heat; lightly oil grate. Grill eggplant, turning occasionally, until skin is charred and flesh is fork-tender, 25–35 minutes. (Alternatively, you can tuck vegetables into coals left over from grilling something else. Wait until charcoal is completely covered with ash and no black spots remain. Shake grill to knock excess ash off coal, then rake them around and pile them up around vegetables.) Let cool slightly.
Halve eggplant, scoop flesh into a colander set over a bowl, and let drain at least 15 minutes and up to 1 hour; discard liquid.
Pulse eggplant along with lemon juice, tahini, and garlic in a food processor until smooth; season with salt and pepper.
Drizzle baba ghanoush with oil and top as desired. Serve with flatbreads or pita bread.