Cut the eggs in half as you would for deviled eggs. Remove the yolks and push them through a fine strainer or mash them with a fork. Mix the egg yolks with the milk, garlic, parsley, salt and pepper. The mixture should be moist and hold together. Stuff the whites with the yolk mixture, reserving approximately 2 tablespoons for the dressing (the egg yolk mixture adds texture to the dressing).
Heat the oil in a large skillet, preferably nonstick. When the oil is hot, add the egg halves, stuffed side down, and fry over medium heat for 2 minutes until browned.
Remove the eggs from the skillet and arrange on a platter.
Put all the ingredients except the oil in a food processor. With the motor running, slowly add the oil.
Pour the dressing on top of and around the eggs and serve.