To make the eggplant puree:
Prepare a gas or charcoal grill for direct cooking over high heat. Brush the grill grate clean. Place the whole eggplant on the grill and char, turning as needed, for about 30 minutes, until all the skin is black and the eggplant is soft all the way through. Transfer to a plate and remove and discard the stem.
While the eggplant grills, in a small pot, combine the garlic and olive oil and place on the grill over a cooler area of the fire. Cook slowly for about 15 minutes, until the garlic is golden and soft. Remove from the grill. Place the onion slices on the grill rack directly over the fire and char, turning as needed, until blackened and soft.
In a small pot, heat the wine over medium-low heat on the stove top until reduced by half. In a blender, combine the eggplant with its charred skin, the garlic and oil, the charred onion, reduced wine, tahini, and 1/2 teaspoon salt and process until a smooth puree forms. Set aside.
To break down the duck:
Working with 1 duck at a time, remove the breasts and then remove the legs, keeping the thighs attached. Save the whole duck legs for the confit on page 286 and use the duck carcass for stock. Trim the breasts and then score the skin side in a crosshatch pattern, cutting at 1/2-inch intervals through the skin and fat but not the flesh. Refrigerate, uncovered, until ready to sear.
To sear the duck and render the fat:
In a 12-inch nonstick sauté pan, heat the rice oil over medium-high for 2 minutes. Season the duck breasts on both sides with salt and carefully place, skin side down, in the pan. Reduce the heat to medium-low and slowly render as much of the fat in the skin as possible, draining the fat that collects in the pan every 2 to 5 minutes. Continue to cook for a total of 10 to 15 minutes, until the skin turns brown and crisps slightly.
Flip the breasts, raise the heat to medium-high, and continue to cook for 2 minutes longer for medium-rare to medium. An instant-read thermometer inserted into the thickest part of a breast should register between 136°F and 140°F. Transfer the breasts to a cutting board and let rest for 4 to 5 minutes before cutting and serving.
To make the salad:
Combine all of the ingredients and stir to mix well.
To make the pickled blackberries:
In a small pot, combine the vinegar, verjus, fish sauce, sugar, and salt and bring just to a boil over medium-high heat, stirring to dissolve the sugar. Remove from the heat, let cool, cover, and chill.
Put the blackberries in a medium bowl and pour the chilled vinegar mixture over the blackberries. Cover and let sit at room temperature for at least 1 hour or up to 2 days. Makes about 1 cup.
To serve:
Cut the duck breasts against the grain into 1/2-inch-thick slices. Divide the eggplant puree between individual plates. Top with a sliced duck breast. Spoon the salad to one side and garnish with pickled blackberries.