While the eggplant is in the oven, place the burrata balls into a large bowl. Using a butter knife, break open the balls and slice until they are well mashed in the bowl. Set aside.
Wash the lemon and zest ½. Reserve the zest—juice the same half of the lemon into a small bowl.
Pit 10 of the black olives (if using salt-cured olives, see note below). Chop fine and set aside.
Chop the cilantro and set aside.
When the eggplant and garlic are cooked, allow them to cool on the counter for a few minutes.
Scoop out the flesh of the eggplant into a bowl large enough to hold all of it, being careful not to destroy the skins of the eggplant if you are going to serve the appetizer in them.
Using the edge of a knife, press on the garlic skin to squeeze out the roasted garlic. Add to the bowl with the roasted eggplant.
Add into the eggplant most of the cilantro, reserving some for garnish. Add in most of the lemon zest, reserving some for garnish. Add in the chili powder and the ground cumin. Add in the lemon juice. Mix well, mashing any large eggplant cubes, but be careful not to overmix.
Place three or four empty eggplant skins into a serving dish. Mound up the eggplant into two of the skins and the mashed burrata into the other skins. Top the burrata with the diced olives and reserved lemon zest. Drizzle olive oil over the burrata. Top the eggplant with reserved cilantro.
Serve warm, cold, or anywhere in between. Garnish the plate with the remaining black olives.