To make the vinaigrette, whisk together the shallot, mustard, vinegar and oil until emulsified. Add salt and pepper to taste and set aside.
To make the salad, bring 3 inches of salted water to a very low simmer—180°—in a medium stainless-steel pan. Crack each duck egg into individual small dishes. Add the distilled vinegar to the simmering water. Gently tilt the eggs out of each dish into the simmering water—keeping the dish as close as possible to the surface of the water.
Allow the eggs to poach for 3 to 5 minutes or until desired doneness. Using a slotted spoon, remove the eggs from the poaching water and drain on a clean towel.
Meanwhile, in a large bowl, toss the mustard greens with the prepared vinaigrette and sprinkle with salt and pepper. Divide the greens onto 4 large plates, top each salad with a poached egg and garnish with the lardons and croutons. Serve immediately.