Trim the ends off of the eggplant and cut it crosswise into 12 1/2-inch thick slices. Season just the tops of the slices lightly with salt.
In a small bowl, whisk together the red wine vinegar, olive oil, dried oregano, garlic, and salt.
Lay the eggplant slices on the grill with the salted side up. Grill, uncovered, for 5 minutes, then flip.
Spoon or brush all of the garlic-herb mixture over the tops, sprinkle evenly with the feta, then cover the grill and continue grilling until the cheese is melted and the eggplant is tender, about 5 minutes more.
Transfer the grilled eggplant to a serving platter and serve hot or warm.