Place the eggs in a saucepan in a single layer. Fill the pan with enough cold water to cover the eggs an inch. Bring to a rolling boil over high heat. Remove from heat, cover, and let stand for 10 minutes.
Drain. Replace water with cold water. Drain again and refill with cold water & ice. Let chill for at least 10 minutes.
Peel the eggs under running water, dry, and then chop into ¼" pieces.
Sauce:
In a medium bowl, whisk together the Mayonnaise, Mustard, Lemon Juice, Worcestershire Sauce, Salt, Pepper, and Sugar.
Add the chopped Eggs, Celery, Scallions, and Parsley. Using a rubber spatula, fold to combine. Taste and adjust seasoning, if necessary.
Serve or refrigerate until ready to use.
Notes
Taste the refrigerated salad again before serving and adjust the seasoning.The eggs can be cooked and stored in their shells in the fridge for a week. Peeled eggs should be stored in an airtight container in the fridge for three days.