Chop garlic; sprinkle with salt and mash into puree with blade of knife held sideways. Scrape garlic-salt mixture into small bowl.
Add oil, oregano and pepper and stir until garlic-salt mixture is well blended.
Preheat gas or charcoal grill to medium.
Cut off and discard thin lengthwise slice from each side of eggplant; cut remaining eggplant lengthwise into about six slices, each about 1/2-inch-thick. Brush both sides of slices with some of oil mixture, scooping up solids and using it all.
Grill until browned on both sides and very tender all the way to center, 5 to 8 minutes per side.
Sprinkle half of eggplant slices with some of cheese, then tomato; top each with another eggplant slice, pressing it into place.
Transfer to serving dish with metal spatula; cut each slice in half crosswise. Serve warm.
Courtesy of Cabot Creamery Cooperative