In a saucepan, bring your water to a boil. Add your eggs gently, and cook, covered for 12 minutes. After 12 minutes have elapsed, remove from the water and into a bowl of ice water to stop the cooking.
Peel and tear your eggs apart into small pieces, or chop them - if you prefer. Place them in a bowl and pour the lemon juice over them. Toss to coat.
Add your mayonnaise, za'atar, salt, pepper, parsley, and onion. Mix thoroughly, and serve over hot toast with thinly sliced Persian cucumbers. Sprinkle with Aleppo pepper, dried harissa, or paprika if desired. Enjoy!