Preheat the grill to 375 degrees F.
Make a saltwater brine, using 4 cups cold water, ½ cup sea salt, 1 tablespoon sumac, 1 tablespoon za’atar and the juice and rind from the ½ lemon. Set aside.
Next, place the eggplant slices in the saltwater brine mix. Let them sit for 30 minutes.
Coat the tomato slices and drizzle with 2 teaspoons of oil and sprinkle with salt and pepper. Place them on the grill, turning quickly to sear both sides, about 3 minutes total. Set aside, keep them warm.
Remove eggplant from the brine and pat dry with a paper towel. Rub with remaining oil and sprinkle with salt, pepper and za’atar. Grill for 5 minutes on each side, until tender but not falling apart. Set aside.
Mix the yogurt, dill, lemon juice, cucumber and mint together and set aside.
Sprinkle burger patties with remaining salt, pepper and sumac on both sides, then grill for 3 minutes on each side for medium rare. (Add 1 minute per side for medium.)
To serve, place eggplant slices on a platter, place a beef patty on each, add a tomato slice and top with the yogurt sauce.