In a bowl, mix the ground lamb with cilantro, lemon zest, garlic, salt, and pepper. Form into two patties.
Brush both sides of the eggplant slices with olive oil and season with salt and pepper.
Preheat your BBQ (or grill pan). Cook the lamb burgers for about 5 minutes per side, or until cooked through. Grill the eggplant at the same time until tender and nicely charred, about 4–5 minutes per side.
Whisk together the Greek yogurt, lemon zest, cilantro, garlic, olive oil, salt, and pepper until smooth.
Toast your buns if you like. Spread the yogurt sauce, add the lamb patty, grilled eggplant, arugula, and feta. Finish with another spoon of sauce if you’re feeling it.