Cut the hard-cooked eggs in half lengthwise. Use your fingers to gently remove the yolks and place them in a medium bowl. Reserve the whites on a medium plate.
Add the mustard, mayonnaise, olive oil, buttermilk, garlic, onion, parsley, lemon juice to the yolks. With a fork, crush everything together into a thick paste. Add salt and pepper to taste.
Spoon 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 which will become the salad dressing.
In a large nonstick skillet, heat the ghee or 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.
To make the salad, in a large bowl combine the leftover egg stuffing with all the dressing ingredients. Whisk to combine and emulsify. 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 enjoy warm.