Preheat the oven broiler on high.
Season the chicken on both sides with salt and pepper. Add the broth, cumin, lemon juice, and chicken to a saucepan or pot covered with a lid. Quickly bring to a boil then lower to a simmer. Cook the chicken to an internal temperature of 160 degrees using a thermometer. Once cooked remove the chicken and shred apart using two forks pulling in opposite directions. Season the broth with salt as needed. Return the chicken to the pot and simmer on low, remove with a slotted spoon when ready to serve.
In the meantime, toss the eggplant with the half of the oil, and season well with salt and pepper. Place on a foil lined baking sheet and under the broiler to roast. (Keep a watchful eye so it doesn't burn). Toss around the eggplant every couple of minutes until soft and browned. Then place in a blender or food processor along with tahini, lemon juice, garlic, cilantro, and remaining oil. Blend until smooth. Add a tablespoon of water if needed to smooth out. Season with salt and pepper to taste.
For the dressing: whisk all the ingredients in a bowl until fully combined. Season with salt and pepper to taste then set aside.
Warm the pita on both sides in a nonstick pan over medium heat.
Spread the babaganoush onto the pita, lettuce, tomato seasoned with salt and pepper, shredded chicken, and sauce. Fold in half and serve.