Preheat the oven to 400 degrees F. Set a cooling rack into a baking sheet.
Halve the eggplant lengthwise and score the flesh by making three long cuts down the middle and sides, making sure not to pierce the skin. Salt the eggplant and then place it
cut-side down on the prepared cooling rack for 30 minutes to release moisture.
Turn the eggplant over and remove the cooling rack. Using paper towels, dry the eggplant and the baking sheet of any moisture. Cut the eggplant into 1-inch cubes and place back on the baking sheet. Scatter the garlic cloves among the eggplant. Drizzle with 3 tablespoons of olive oil. Sprinkle with the smoked paprika and some salt and pepper. Toss to coat. Roast until the eggplant is completely soft and browned, about 30 minutes. Let cool.
Increase the oven temperature to 425 degrees F.
Lightly oil a baking sheet. Toss the potatoes in a medium bowl with 1 tablespoon olive oil, 1/2 teaspoon salt and some pepper. Place the potatoes cut-side down on the oiled baking sheet and roast until tender, 13 to 15 minutes. Let cool.
Scrape the cooled eggplant into the bowl of a food processor and squeeze the roasted garlic out of its peels into the food processor. Add the lemon zest, lemon juice, parsley, mint and the remaining 2 tablespoons olive oil and puree until smooth. Add the shallot and tomato and pulse to combine. Season with salt and pepper, if necessary.
To serve, scoop out a cavity in each potato half with a paring knife or small spoon. Fill with the "eggplant caviar" and top with a dollop of creme fraiche and a sprinkle of black sesame seeds.
This recipe yields a little extra eggplant caviar, which can be served as a dip with pita bread or breadsticks. It would also be good as a sandwich spread.