Sprinkle both sides of the eggplant slices lightly with salt and place between layers of paper towels. Let stand for 1 to 2 hours, rinse, and pat dry.
Combine the olive oil, vinegar, garlic, oregano, salt, and pepper. Place the eggplant slices on a grill over moderate heat and baste with the oil mixture. Turn and baste frequently until lightly browned, about 8 minutes.
Top each slice with a slice of cheese and grill an additional 2 minutes, until the cheese begins to melt.