Seed and chop the cucumber. In a medium bowl, stir together the cucumber, yogurt, garlic, dill and feta. Refrigerate and chill for 1 hour.
In a small bowl, combine vinegar, oregano, salt and pepper. Whisk in oil until thick and well blended. Cut the eggplant into 1/2-inch rounds. Arrange in a 13x9x2-inch baking dish; coat well with dressing. Grill eggplant until tender, about 3 minutes each side, keep warm.
In a large bowl blend together lamb, mint, cumin, salt and pepper, to taste. Form 6 patties.
Before grilling burgers, brush the grill rack with oil. Place the patties on the rack turning once, 4 minutes each side.
Cut pitas horizontally and lightly toast over indirect heat.
Assemble burgers as follows: pita bottom, lettuce, burger topped generously with tzatziki, a slice of grilled eggplant, and the top of the pita.