Light the grill.

Grill the eggplants whole over medium flame on the barbecue. Turn so that they char and soften all around. They will need between 15 and 30 minutes to grill, soften and cook through, depending on how large and firm they are. When the eggplants are done, remove with tongs to a large stainless steel bowl and seal either with plastic wrap, aluminum foil or a close-fitting lid. Let the eggplants cool inside the bowl until you can handle them.

Place ½ cup olive oil inside a large mixing bowl. Remove the eggplants one at a time from their bowl and transfer to a cutting board. Score down the middle lengthwise to open into two halves, like a butterfly. Score the flesh with a paring knife and remove it with a tablespoon, scraping as much out as possible. Place the warm eggplant flesh directly in the olive oil and pick out any pieces of charred skin that may fall in. Continue with remaining eggplants.

Season to taste with salt and a pinch of sugar. Stir the eggplant pulp and olive oil together, adding more olive oil, if needed, to make a smooth and somewhat velvety but chunky mash. Season with garlic and/or balsamic if desired, and stir in the parsley. Set aside.

Toss the cleaned anchovies with olive oil, salt and pepper and grill lightly, turning once. They should take just a few minutes to cook through.