Heat a grill to medium. Cut the eggplant in half lengthwise and brush the cut sides with 2 tablespoons of the olive oil. Grill, turning the pieces as necessary, until cooked through and soft. Cool, then scrape the flesh into the bowl of a food processor. Discard the skin.
While the eggplant cooks, soak the tomatoes in boiling water for 15 minutes. Drain and chop. Add to the processor with the eggplant. Add the remaining ingredients, including the remaining 2 tablespoons of olive oil, and process until smooth. Taste and adjust seasonings, as desired.
Serve immediately or refrigerate in an airtight container for up to 4 days.