Mash together the egg yolks, milk, garlic, parsley. Add salt and pepper to taste. Mixture should be moist, but hold together.
Stuff egg whites with yolk mixture, reserving about 2 Tbs. Do not mound yolk mixture too high in the whites. Set aside.
Stir together the reserved egg yolk mixture, mustard, water and salt and pepper in a bowl.
Whisk in olive oil until it is a nice salad dressing consistency.
Heat olive oil and butter in non-stick frying pan. When they foam up, add eggs, stuffed side down. Fry for about 2 minutes until the yolk mixture is nice and brown. The whites will not brown--just get rubbery if you overcook them.
When done, remove to a serving plate and drizzle the dressing over. Serve warm.