2. Whisk together the 1/2 cup oil, vinegar, parsley, garlic and 1/4 teaspoon each salt and pepper until combined. Set aside.
3. Toss the chicken with 2 tablespoon of the oil and the lemon juice and sprinkle all over with salt and pepper. Brush both sides of the eggplant slices with the remaining 4 tablespoons oil. Sprinkle salt and pepper on both sides.
4. Grill the chicken, skin-side down, until the skin is crisp and lightly charred, about 12 minutes. Flip and grill until cooked through, about 15 more minutes. Meanwhile, grill the eggplant, flipping once, until tender and lightly charred on both sides, about 10 minutes total. Grill the haloumi, turning once, until soft and lightly grilled on both sides, about 5 minutes total. Cut both the eggplant and the cheese into 1/2-inch pieces.