Bring 1 inch water to rolling boil in medium saucepan over high heat.
Take 6 cold large eggs from the refrigerator and put them in a steamer basket. Transfer basket to saucepan.
NOTE: If you don’t have a steamer basket, use a spoon or tongs to gently place the eggs in the water. It does not matter if the eggs are above the water or partially submerged.
Cover, reduce heat to medium (small wisps of steam should escape from beneath lid), and cook eggs for 13 minutes.
When eggs are almost finished cooking, combine 2 cups ice cubes and 2 cups cold water in 2nd saucepan.
Using tongs or spoon, transfer eggs to ice bath; let sit for 15 minutes.
Peel eggs and slice each in half lengthwise with a paring knife. Remove yolks to a small bowl.
Arrange whites on a serving platter or plate, discarding two worst-looking halves.
Mash yolks with a fork until no large lumps remain.
Add mayonnaise, mustard, and vinegar and mix with a rubber spatula, mashing the mixture against the side of the bowl until smooth. Salt and pepper to taste.
Put the mixture into a zip-lock bag and cut off one bottom corner tip of the bag (or use a pastry bag with a large open star tip).
Twist the top of the bag to help push the mixture toward the open tip of the bag (the tip you cut off).
Pipe the yolk mixture into egg white halves, mounding filling about 1/2 inch above the flat surface of the whites.