Place the eggplant in a colander, salting a little at a time. Let stand in the sink for 1 hour to drain. Rinse and pat dry with paper towels.

Season the chicken with salt and pepper to taste. Heat 2 tablespoons olive oil in a large, wide, nonstick pot or Dutch oven over high heat, add the chicken breasts, and saute, turning until browned on both sides, about 8 to 10 minutes. Remove the chicken to a platter and set aside.

Using the same skillet, reduce the heat to medium, add the onion and celery, and saute until translucent, about 5 to 6 minute. Add 3 tablespoons olive oil, raise the heat, and add the eggplant. Saute until the eggplant is lightly browned, 4 to 5 minutes. Add the garlic, stir gently for 1 minute, and add the tomatoes.

Transfer the chicken back to pot, add the oregano, cover, lower the heat, and cook for 25 minutes, until the chicken is tender. Just before serving, sprinkle in the feta and parsley.