Turn all burners on high and heat for about 15 minutes, until very hot. Clean cooking grates. Lay whole eggplants directly on grates over high heat. Cook until all sides are soft when pressed, about 25 minutes, turning occasionally.
Grilled Eggplant - Oven Method
Move oven rack to middle position and heat oven to 500°F. Line a rimmed baking sheet with foil. Roast eggplants on the baking sheet until soft when pressed, about 60 minutes, turning every 15 minutes.
Baba Ghanouj
Transfer the roasted eggplants to a large plate or rimmed baking sheet until cool enough to handle (5-10 minutes). Place a colander in the sink or strainer over a bowl. Cut the tops and bottoms off the eggplants, and cut in half lengthwise. Scoop flesh with a spoon from the skin and set in the colander or strainer for 3 minutes to drain. Discard skin.
Transfer drained eggplant to a food processor and add tahini, lemon juice, garlic, salt, pepper and smoked paprika. Process until smooth, about 30 seconds. Transfer to a serving bowl and cover the surface directly with plastic wrap. Chill 45-60 minutes before serving. To serve, drizzle top with olive oil and sprinkle with parsley and an additional pinch of smoked paprika.
Notes
Tahini is a Middle Eastern sesame seed paste. Find it in the ethnic food or natural foods section, or possibly with the nut butters.