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Peel and halve the eggs. Remove the yolks and place them in a bowl.
Add garlic, chopped parsley, a squeeze of lemon juice and milk to the egg yolks. Mix until smooth, then season to taste with salt and pepper.
Spoon the mixture into the egg whites, flattening the yolks into the whites.
Make the vinaigrette by whisking the leftover egg yolk mixture, mustard, lemon juice, water, and olive oil until thick and smooth. Season with salt.
In a nonstick pan, heat the olive oil over medium. Add the eggs face down and cook until golden brown, about 2 minutes.
Spread the vinaigrette on a platter and top with the eggs, or arrange the eggs on the platter and spoon the vinaigrette on top.