Israeli Eggplant and Egg Sandwich (Sabich)

Ingredients

• Peel of 1 large yellow onion
• 4 hard-boiled eggs, peeled
• 7 tbsp. canola oil
• 1 1⁄2 lb. large eggplant, cut crosswise into 1/4"-thick slices
• Kosher salt
• freshly ground black pepper
• 2 Kirby cucumbers, unpeeled, finely chopped
• 1 small tomato, cored and finely chopped
• 1 small red onion, finely chopped
• 3 tbsp. fresh lemon juice
• 2 tbsp. extra-virgin olive oil
• 5 tbsp. tahini
• 1 clove garlic, finely chopped and mashed into a paste with a little salt
• 4 pitas, warmed
• Amba, for serving
• 1⁄4 cup packed flat-leaf parsley leaves