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.
Meanwhile, 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.