Whisk the vinegar, mint, red pepper flakes, 1 clove garlic and 1/4 teaspoon salt in a shallow dish. Gradually whisk in 3 tablespoons olive oil. Add the chicken and marinate about 15 minutes.
Meanwhile, mix the tomatoes, cucumber and the remaining 1 clove garlic in a bowl. Drizzle with olive oil and season with salt and pepper.
Preheat a grill pan over medium-high heat. Grill the chicken until cooked through, 2 to 3 minutes per side. Transfer to a cutting board and slice into 1/2-inch-thick strips. Toss with the vegetables.
Brush the pitas with the remaining 1 tablespoon olive oil and season with salt. Grill, turning once, until marked. Place a pita on each plate and spread with baba ghanoush. Top with lettuce and chicken salad and drizzle with any juices from the bowl.
Photograph by Antonis Achilleos
Recipe courtesy Food Network Magazine