Bring water to a boil with a little vinegar. Add the eggs one at a time using a tbsp to prevent them from cracking. Cook for 10 minutes.
Remove any wilted leaves from the leeks. Trim off the root end, then cut the leeks lengthwise. Rinse the leeks under running water, taking care to spread the layers apart and remove any dirt from inside.
Cut the leeks into pieces about 10 cm long. Steam them for 15 minutes.
If you’re using the steamer basket of a pressure cooker, you can reduce the cooking time to 10 minutes once the valve starts hissing.
Keep the leeks warm and covered.
While the leeks are cooking, peel, remove the roots, and finely chop the shallots. Mix the oil and vinegar in a bowl. Add the shallots.
Crack the eggs and separate the whites from the yolks. Add the yolks to the bowl and set the whites aside.
Mash the egg yolk and mix it thoroughly into the vinaigrette until smooth. Cut the egg white into pieces and add it to the vinaigrette.
Just before serving, cut the leeks into equal pieces about 10 cm long, if you haven’t already done so. Place the warm leek pieces on each plate. Drizzle with the egg vinaigrette and serve immediately!