For tahini dressing, in a small bowl whisk together 1/4 cup of the oil, the tahini, lemon juice, garlic, and bottled hot pepper sauce. Season to taste with salt and black pepper. Cover and chill until ready to serve.
Brush eggplant and pita bread with the remaining 2 tablespoons oil. Sprinkle with paprika, garlic powder, and additional salt and black pepper.
For a charcoal or gas grill, place eggplant slices on the rack of a covered grill directly over medium heat. Grill for 8 to 12 minutes or until very tender and well browned. Remove eggplant from grill. Place pita bread on grill rack. Grill for 2 to 4 minutes or until toasted, turning once halfway through grilling. Cut eggplant into chunks. Cut pita bread into wedges.
Line a serving platter with mixed greens; sprinkle with mint and parsley. Arrange eggplant, pita wedges, cucumber, and garbanzo beans in piles on top of greens. If desired, sprinkle with pistachios. Drizzle with tahini dressing.