Prepare a grill for medium-high heat; lightly oil grate. Grill beets, turning occasionally, until skin is charred and flesh is fork-tender, 50–60 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 beets; scoop flesh into a food processor. Add lemon juice, tahini, and garlic and process until smooth; season with salt and pepper.
Drizzle baba ghanoush with oil and top as desired. Serve with flatbreads or pita bread.
Do Ahead: Dip can be made 3 days ahead. Cover and chill.