The origins of leeks vinaigrette—poached leeks in a mustardy dressing—are unknown, but it's easy to imagine someone pulling them out of the stockpot once they had worked their magic, then seasoning them.
Starting about 1" above root end, slice leeks lengthwise, but not all the way through. Open leeks like a book and wash well in cold running water to remove all sand and dirt. Bring a 12" deep-sided skillet of salted water to a boil, add leeks, and cook over medium heat until soft but not mushy, about 6 minutes. Transfer leeks to a large bowl of ice water to stop them from cooking further. Carefully split leeks completely in half lengthwise, and transfer to a rack, cut side down, to drain thoroughly.
Whisk vinegar, mustard, and salt and pepper together in a small bowl. Gradually add oil, whisking constantly, until vinaigrette is smooth and creamy. Adjust seasonings and set aside. Remove leaves from 4 of the parsley sprigs, chop leaves, and set aside.
Divide leek halves equally among 4 warm salad plates. Drizzle vinaigrette over leeks, and sprinkle with chopped parsley and egg. Garnish each plate with a sprig of parsley.