This flavorful dish can serve as a vegetarian main course or a side dish to any grilled or roasted meat. Serves 4.
Prepare the grill so that the temperature on 1 side is hot and the other is just medium-low (you should be able to hold your hand 5 inches about the grill for about 5 seconds).
Whisk together the vinegar, rosemary, and salt and pepper, to taste. Slowly whisk in the olive oil.
Peel the eggplant and cut crosswise into 1/3-inch slices (12 slices total). Cut the tomatoes crosswise into 1/3-inch slices (8 slices total). Place the vegetables on a tray and brush generously with the dressing.
On the grill's hot side, grill the eggplant until just cooked through, 2 to 3 minutes per side, brushing with more of the dressing. Grill the tomato slices for 1 minute per side. (The vegetables should retain their shape.)
Transfer the vegetables to a tray as they finish grilling. Arrange 4 stacks of vegetables on the cooler side of the grill: start with the eggplant, add a slice of tomato, then add some crumbled cheese. Repeat. Cover the grill and cook the stacks or 3 to 5 more minutes or until the feta melts. Serve immediately.