Place the bulgur in a medium-sized bowl and add the water. Toss it a little and let it sit, covered with a cloth, for 2 to 3 hours, until all the water is absorbed. (You can hasten this process by following package directions and adding boiling water to the bulgur instead. Whatever method you choose, the ratio of water to bulgur should always be 2:1, to ensure that the grain stays toothsome and fluffy.)
Wash and pat dry the eggplant. Trim the stem and bottom. Cut the eggplant in half lengthwise, and cut each half into four or five strips. Cut each strip into small cubes, a little less than an inch square. Place them in a colander, sprinkle with 1 to 2 teaspoons salt. Place a weight (such as a pot cover) over the eggplant and let drain for 30 minutes. Rinse and drain well, squeezing to remove excess liquid, and pat dry.
Heat half the oil in a large skillet and add the diced eggplant (you may have to do this in two batches). Stir-fry the eggplant continuously until it is coated with the oil and begins to soften. Add the garlic and continue to fry until the eggplant is very soft, 8 to 10 minutes.
Remove from heat and toss together with the bulgur. Add the dill and mint to the bulgur salad and toss to combine.