Position one rack in the center of the oven and another rack 4 to 5 inches from the heat source. Preheat the oven to 400°F. Lightly brush a 13 x 18 x 1 inch sheet pan with a little of the olive oil.

Place the eggplant, flesh side up, on a work surface or large tray. Season generously with salt (about 1½ teaspoons) and set aside to 'sweat' for about 20 minutes while the oven is preheating.

Using paper towels, pat the eggplant dry, removing excess water and salt. Arrange the eggplant in the sheet pan, flesh side facing up. Brush generously with about 3 tablespoons of the olive oil (or more if needed to coat all the pieces), and roast for 25 to 30 minutes, until fork-tender. For a little more color (which I highly recommend), set the oven to broil and place the eggplant on the top rack under the heat source for the last 1 to 2 minutes, watching carefully (it should gain a nice deep, golden brown color but it should not burn).

While the eggplant roasts, pound the garlic and jalapenos into a chunky paste for the dressing using a mortar and pestle. (Or, mince the garlic and jalapeno as finely as you can and transfer to a medium bowl.) Add the capers, lime juice, vinegar, and 3 tablespoons of olive oil; whisk to combine.

As soon as the eggplant is cooked through, carefully transfer the pieces to a large bowl. Sprinkle with the sumac, then pour the vinaigrette over the eggplant and add the fresh mint. Using a pair of tongs, give the hot eggplant a gentle toss to coat. Set aside to marinate for 20 minutes, then serve at room temperature.