In a small bowl, combine the vinegar, oregano, salt and pepper and then whisk in the olive oil until the mixture is thick and well-blended.
Arrange the eggplant slices in a 9x13-inch baking dish. Pour the dressing over the slices and turn the eggplant to coat well. Set aside.
MAKE THE BURGERS:
Preheat a gas grill to medium high.
In a large bowl, combine the lamb, mint, and cumin and season with salt and pepper. Handle the meat as little as possible- you just want to mix the flavors.
Divide the mixture into 6 equal portions and form the portions into patties to fit the pita breads.
When the grill is ready, brush the grill rack with vegetable oil. Place the patties on the rack, cover, and cook, turning once, just until done, about 4 minutes on each side for medium-rare. Remove from heat.
Grill the eggplant slices until tender, about 3 minutes on each side.
To assemble- in each pita add ½ cup lettuce, a patty, about ¼ cup of tzatziki sauce, and an eggplant slice.
Notes
If you are serving this as gluten-free, serve on GF bun or bread, or serve without a bun.