Step 1Peel the eggplant and cut into 1-inch chunks. Toss with the coarse salt. Place in a sieve over a bowl and let stand for at least 1 hour.
Step 2Rinse the eggplant well. Working in batches, gently squeeze out as much moisture as possible between your hands.
Step 3In a large skillet, preferably nonstick, heat the duck fat over moderate heat. Add the eggplant, cover with a lid, and cook, turning the pieces from time to time, until they begin to plump up, feel tender, and turn golden brown on all sides, about 10 minutes.
Step 4Reduce the heat to low, add the sugar, and slowly cook, uncovered, turning the pieces of the eggplant often, for another 5 to 8 minutes, until tender and glazed. Remove the skillet from the heat. (The recipe can be prepared to this point up to 2 hours in advance.)
Step 5Shortly before serving, add the parsley and garlic to the eggplant. Toss to mix. Cover and cook over low heat to rewarm. Season with salt and pepper to taste and serve.