Serve this appetizer with Yukon Gold Potato Blini from chef Thomas Keller's "The French Laundry Cookbook."
Halve eggplant lengthwise; score the flesh side of both halves, making a crisscross pattern about 1/4 inch deep. Sprinkle both halves with salt and place scored-side down on a rimmed baking sheet. Cover with a second baking sheet and place a heavy object (such as cans or a set of bowls) on top. Let eggplant render its excess liquid at room temperature for 1 to 2 hours.
Preheat oven to 350 degrees. Lightly oil a baking sheet and set aside.
Rinse eggplant, pat dry, and rub with some olive oil. Place halves flesh side down on prepared baking sheet. Transfer to oven and roast until eggplant is very soft and skin is wrinkled, about 1 hour. There should be no resistance when tested with a knife. Remove from oven and let stand until cool enough to handle.
Scoop out eggplant pulp and lightly chop. Place pulp in a double layer of cheesecloth and tie to enclose with kitchen twine, leaving about 6 inches on the end. Hang bundle from a shelf in the refrigerator with a bowl set underneath or place in a colander set over a bowl and refrigerate; let drain at least 2 hours and up to overnight.
Remove eggplant from refrigerator and hold cheesecloth bundle over sink; twist and squeeze to remove as much additional liquid as possible. Transfer pulp to the bowl of a small food processor; pulse a few times to break up eggplant. With the motor running, slowly add 2 tablespoons olive oil through the feed tube; continue processing until the color and texture begin to lighten, about 30 seconds.
Turn the machine off and add garlic, mustard, and salt. With the motor running, slowly add remaining 2 tablespoons olive oil; continue processing 1 minute more. Season with salt and pepper. Eggplant caviar may be kept refrigerated in an airtight container for up to 1 week.