Make the sauce: add all the ingredients to a blender / food processor container. Mix until the sauce is smooth. Season to taste with salt and pepper.
Cook the eggs: Add the eggs into a small pot and pour the boiling water over. Bring the water to a boil and cook the eggs for about 5 minutes and 30 seconds over high heat (my favorite method). Add the eggs into cold water and peel them.
Make the tomato cucumber salad: Cut the tomato and cucumber into small cubes (you can peel them if you wish). Toss with the olive oil, lemon juice, and chopped parsley. Season with salt and pepper to taste.
Cook the eggplant: Cut the eggplant into 1/2-inch (1 cm) slices. Heat the oil in a big frying pan and cook the eggplant in batches until it's browned on both sides and soft. Try not to overcrowd the eggplant slices in the pan, letting the eggplant brown nicely. Transfer to a plate lined with paper towels and season immediately with salt and pepper to taste (while the eggplant is still hot).
Warm up the pita bread: You can do that on a dry pan or panini grill. Cook the pita bread until warm and lightly golden in color.
Assemble the sandwich: Open the pita bread, spread a heaped tablespoon of the sauce on both sides of the pita bread, add 2-3 eggplant slices, 2 tablespoons of the salad and 1 cut in half egg.
Enjoy!