Bring a pot of water to a boil (I place the eggs in the water while it is still cold; I find that if I put cold eggs in boiling water, they are more likely to crack once they go in the hot water).
Boil the eggs for 10 - 15 minutes, or until how done you like your eggs.
Place the cooked eggs in a bowl with ice water to begin cooling them down.
Under cold running water, crack each egg all the way around and gently lift the shells and membrane off . . . get under the membrane for the easiest removal of the shell. This works like a dream each and every time!
Slice each egg in half.
Gently scoop out the yolk and place in a medium bowl.
Place the white portion of the eggs on a plate or egg platter.
Mash the yolks with a fork.
Add the mayo, mustard, vinegar, and melted butter, salt and pepper.
Whip with a fork until smooth.
Taste and make any adjustments that you prefer.
Spoon heaping amounts of this egg mixture into the cooked whites.
Sprinkle with paprika.
Garnish with fresh-cut herbs.
Cover and chill in the refrigerator (I try to always make these in the morning so that they are cold for lunch and/or later on in the day).