Trim the roots from each leek but leave the root end intact. Cut off the dark tops so that you have only white and pale green parts (save the tops for stock). Cut the leeks in half lengthwise, being careful not to cut through the root ends. Wash the leeks thoroughly under cold water, checking for dirt between the layers of leaves.
Bring a pot of water with a steamer insert to a boil. Cook the leeks until tender, 10 to 15 minutes. (They can also be boiled if you don't have a steamer.) Remove the insert from the pot and run the leeks under cold water to cool them, then put them on a plate lined with paper towels/absorbent paper to drain. Refrigerate until you are ready to serve them.
Put the vinegar, mustard, and honey in a blender. Add a two-finger pinch of salt and several grinds of pepper. With the blender running, pour in the oil in a steady stream. Transfer the vinaigrette to a glass measuring cup. About 10 minutes before serving, stir in the shallots.
Cut the root end off each leek. Arrange the leeks on plates and spoon the vinaigrette over the leeks. Garnish each plate with chopped egg white, followed by the chopped egg yolk and chives.