Rub the eggplant and onion with the olive oil. Place on a baking sheet. Bake until the eggplant and onion have both softened, turning once, about 25 to 30 minutes. Add the garlic to the baking sheet with the eggplant and onions at the halfway point, after about 15 minutes. Cool until warm enough to handle.
Peel the eggplant and the garlic and place in the bowl of a food processor. Cut the onion into quarters or large chunks, and add to food processor. Add vinegar, lemon juice, Parmesan, salt, and pepper. Pulse until combined. Pulse longer if making a dip. Add cilantro. Taste and adjust seasonings. Refrigerate until ready to eat.