Bring a large pot of water to a vigorous boil and season liberally with salt.
Trim the leeks at the root ends, keeping the leeks intact. Cut the leeks where the light green meets the darker green. Save the darker green section for stock. Split the leeks in half lengthwise, again leaving the roots intact. Put the leeks in a large bowl of water and let it sit for a minute. Rinse the dirt from the insides of the leeks by lightly separating the layers and allowing the water to wash the dirt out. Tie the leeks in 3 places with butcher's twine to hold shut.
Add the leeks to the pot of water and let it come back to a simmer. Boil the leeks until they are very tender, 10 to 12 minutes. Transfer them to paper towels and remove the twine. Separate each leek into 2 halves.
In a small bowl, combine the oil, vinegar, mustards, garlic, shallots and salt and pepper. Spread some of the vinaigrette on the bottom of an 8-inch square dish.
While the leeks are still warm, remove the outer layer of each; lay the leeks cut-side down in the vinaigrette in the dish. Spoon the remaining vinaigrette over the leeks. Cover and chill the leeks in the refrigerator for at least 3 hours and up to overnight.
Arrange the leeks on a platter and drizzle with the vinaigrette. Garnish with the hazelnuts, eggs and chives. Drizzle more vinaigrette over the leeks and serve.