Position an oven rack 6–8 inches below the broiler and preheat to high. Line a rimmed baking sheet with foil. Prick each eggplant several times with a fork to vent.
Broil on the prepared sheet, turning every 5–7 minutes, until the skins are deeply blackened and the flesh is very soft and collapsed, 25–35 minutes. (Alternatively, grill over medium-high direct heat, turning occasionally, 15–25 minutes, to the same doneness.)
Transfer to a bowl, cover loosely, and let steam until just cool enough to handle, 10–15 minutes. Split, scoop out the flesh, and discard the skins and any very seedy pockets.
Drain the flesh in a fine-mesh strainer or colander for 10–15 minutes to shed excess liquid, then transfer to a bowl and mash with a fork until slightly chunky.
Stir in the tahini, lemon juice, garlic, and kosher salt until creamy and well combined, 1–2 minutes.
Transfer to a shallow bowl, swirl the surface, drizzle with extra-virgin olive oil, and sprinkle with chopped parsley. Serve at room temperature or chilled.