Cut the eggs in half crosswise at the widest point. Remove the 1 yolks and push them through a food mill fitted with the small screen. If you do not have a food mill, mash the yolks by hand, but do not use a food processor, which would make the yolks gummy and rubbery. Mix the sieved egg yolks with the milk, garlic, parsley, salt, and pepper. The mixture should be moist and hold together. Restuff the whites with the yolk mixture, reserving approximately 2 tablespoons for the sauce.

Heat the oil and butter in a skillet, preferably the nonstick type. When the oil and butter are hot and foaming, place the egg halves, stuffed side down, in the skillet. Fry at medium heat for about 2 minutes. They will brown beautifully on the stuffed side. (Egg whites do not brown well and get tough if cooked in the hot fat.) Remove the eggs from the skillet, and arrange them over the sauce in a platter or gratin dish. Serve lukewarm or at room temperature with crusty French bread.

The 2 tablespoons of leftover egg-yolk mixture add texture to the dressing, which can be used like any thin, mustardy mayonnaise. Substitute 1 boiled egg for the egg-yolk mixture if the sauce is made for something other than our recipe.

Place the first 6 ingredients in a food processor, and, with the motor on, add the oil slowly. Pour the dressing into a serving platter or gratin dish, and arrange the egg halves on top.

Immerse the eggs in cold or lukewarm tap water, then bring to a boil (for 6 eggs this should take about 2 minutes). Simmer gently. If the water boils too fast, the egg whites tend to get rubbery. After 8 minutes plunge the eggs into cold water to stop further cooking and to prevent the yolks from turning greenish-brown. Keep under running cold water until thoroughly cold. The yolks should still be a bit soft in the center.