Bring a small sauce pot of water to a boil and fill a small bowl with ice water. 

• Remove the stem from the eggplant; cut the eggplant in half lengthwise, then crosswise into ¼-inch-thick half-moons. Sprinkle the half-moons with salt and lay them on a paper-towel-lined sheet pan. Let stand for 10 minutes; pat dry.

Carefully lower the eggs into the boiling water and cook for 10 to 12 minutes for hard-cooked. Transfer the eggs to the ice water to cool. Carefully peel the eggs, then coarsely chop. While the eggplant is standing and the eggs are cooking, prepare the salad.

• Peel the cucumber, if desired; cut the cucumber in half lengthwise, then crosswise into enough ¼-inch-thick half-moons to measure 1½ cups [3 cups]. 
• Cut away the cores from the tomatoes; cut the tomatoes into quarters.
• Strip the parsley leaves from the stems; coarsely chop the leaves.
• Juice the lemon. [Juice 1 lemon; save the remaining lemon for another use.]

In a medium bowl, toss together the cucumber, tomatoes, parsley, 1 tablespoon [2 TBL] lemon juice, and 1 teaspoon [2 tsp] oil. Season to taste with salt and pepper. 

In a large frying pan over medium-high heat, warm 1 to 2 tablespoons oil until hot but not smoking. Working in batches if needed, add the eggplant, season with salt and pepper, and cook, turning once, until lightly browned and tender, 3 to 5 minutes. Transfer to a plate. Add more oil between batches if needed. While the eggplant is cooking, toast the pita breads. 

On the stovetop directly over a flame, or in a dry medium frying pan over medium heat, warm the pita breads, turning once, until lightly toasted, about 30 seconds per side. Alternatively, toast them in a toaster oven. Cut in half. 

Divide the pita halves between individual plates and spread 1 tablespoon hummus inside each pita half. Fill the pitas with the eggplant, cucumber salad, eggs, and as much mango amba as you like. Serve any remaining fillings on the side.