Cut the hard-cooked eggs in half lengthwise. Gently remove the yolks (fingers work best), and place them in a medium bowl. Reserve the whites.

Add the mustard, garlic, onion, parsley, milk, mayonnaise and vinegar to the yolks. With a fork, crush everything together into a thick paste. Add salt and pepper to taste.

Pack the mixture back into the hollows of the egg whites, so the filling is even with the surface of the egg, not mounded. You will have leftover stuffing (this becomes the salad dressing).

In a large nonstick skillet, heat the oil over medium heat. Gently place the eggs in the pan, stuffed side down. Cook until the eggs are beautifully browned, 3 to 5 minutes. Sprinkle them with salt and pepper as they cook.

As the eggs sauté, combine in a large bowl the leftover egg stuffing with all the dressing ingredients. Add the salad greens to the bowl, and toss. Heap them on a serving platter.

Gently lift the eggs from the pan, turn them filling side up, set them on the greens, and serve.