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.