How to Make It
Preheat the broiler and set a rack 6 inches from the heat. Cut 1 leek in half lengthwise. Separate the layers and arrange on a foil-lined baking sheet. Broil, turning occasionally, until burnt and crisp, 10 to 15 minutes; transfer the pieces to a plate as they char. Let cool, then crumble the pieces to make the leek ash.
Meanwhile, tie the remaining leeks into 2 bundles of 4 with kitchen twine. Cook in a pot of unsalted boiling water until tender, 8 to 10 minutes. Cool the leeks in a bowl of ice water. Discard the twine and pat the leeks dry. Cut into 1/2-inch rounds and blot dry.
In a large bowl, combine 3 tablespoons of the olive oil, 2 tablespoons each of the vinegar and orange juice and the orange zest. Add the leeks and turn to coat; season with salt and pepper.
Cook the eggs in boiling water for 6 minutes, then transfer them to a bowl of ice water to cool, about 3 minutes. Peel the eggs and carefully separate the whites from the runny yolks. Chop the egg whites and transfer to a small bowl. Stir in the shallot, mint and 1 teaspoon of the vinegar and season with salt and pepper.
In a small bowl, combine the egg yolks with 1 tablespoon of the oil and the remaining 2 tablespoons of orange juice. Season with salt and whisk until smooth. Strain into a small bowl. In another bowl, combine both of the mustards with 2 tablespoons of the oil and the remaining 1 tablespoon of vinegar; season with salt and pepper.
In a blender, puree the parsley with the remaining 1/4 cup of oil until smooth.
Spread some of the mustard vinaigrette, parsley puree and egg yolk sauce on plates. Top with the leeks vinaigrette and the egg whites. Garnish with leek ash and serve.