1 tablespoon white vinegar

1 tablespoon sugar

1/4 teaspoon of salt, or to taste

1 tablespoon yellow mustard, or to taste

black pepper to taste

2 - 3 heaping tablespoons of mayonnaise

paprika to sprinkle if desired

Place eggs in a large saucepan and just cover with water. Place on stove, cover with a lid and bring to a boil. Boil for 5 minutes, then turn off heat and leave on the burner, covered, for 15 minutes.

Remove lid and set aside. Pour off hot water and run cold water over eggs, pouring off water several times. While eggs are still warm to the touch, pour off all water except for about 1 inch in the bottom of the pan, then toss in a handful or 2 of ice cubes. Place the lid back on the pan, grasp pot handle and lid firmly, and shake vigorously several times. You'll hear the eggs and ice colliding and cracking the shells. 

Now remove the lid and run more cold water over the eggs. They should be cracked and the shells should be very easy to finish removing now. Some may have slipped from their shells entirely.

Keep the cold water running over the eggs as you peel them, and by the time you finish peeling all the eggs they should be room temp or fairly close. 

Cut the eggs open lengthwise as shown in Jessica's photo above, and place the egg yolks in a medium sized bowl. Place the eggs on a platter or other serving dish. 

Mash the yolks with a potato masher or a fork, making sure all the lumps are mashed out and they have a fine consistency. Add the vinegar, sugar, mustard, salt, and pepper, mixing well. Add mayo until it reaches a consistency that pleases you. Start with low amounts, because you can always add more, but you don't want it so runny it won't stay in the eggs. 

Either spoon filling into each egg half, or you may place the filling in a zip-top bag and snip a small hole in one corner and squeeze to fill each egg. Sprinkle prepared eggs with paprika if desired.

Chill until serving time, and keep any leftovers refrigerated. Recipe may be increased as needed.