Preheat the grill to high. Brush the lavash with oil on both sides. Grill until golden brown on both sides. Brush the eggplant with oil and season with salt and pepper. Grill the eggplant, cut-side down, until golden brown. Turn the eggplant over and grill until just cooked through. Remove from grill and cut into 1/4-inch thick slices.
Spread the hummus over the bread, top with the eggplant, and then the feta. Grill, with the cover down for 2 to 3 minutes, or until cheese is melted. Remove from the grill, drizzle with the red chile oil and sprinkle with mint.
Put the chickpeas, garlic, and tahini in the work bowl a food processor. Pulse until it becomes paste-like, adding a few tablespoons of water, if needed. Add the cayenne, cumin, and lemon juice, and pulse again until combined. With the motor running, slowly add the olive oil until emulsified. Add the parsley and pulse until just combined. Season with salt and pepper, to taste. Transfer to a small bowl.
Place all ingredients in a blender and blend for 5 minutes. Strain into a small bowl.