Preheat the Indoor/Outdoor Grill on setting 3.
In a small bowl, mix together the balsamic vinegar and brown sugar. Set aside.
Salt both sides of the eggplant slices and let sit for 5 minutes. Pat dry with a paper towel.
Dip the slices in the balsamic mixture and place on the preheated grill. Place the lid on the grill.
Grill the eggplant slices for 4 minutes per side.
Serve with rice or pasta.