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Put the eggs in a single layer in a pot and cover with cold water to an inch and a half over the eggs. Let the eggs boil on a medium high heat for 7-10 minutes. After 7-10 minutes, move the pot from the burner on the side, and let the eggs stay in the pot covered for the next 5 minutes. Drain the hot water, and replace it with cold. Try to cool down the eggs in cold water for a few minutes, changing the water more than several times. Once cool enough to the touch, peel the eggs under cold running water. *Often I break the egg shell and drop back in the cold water, leaving them for a few minutes for easier peeling.
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Slice each egg in half lengthwise with a knife, starting from the round end and go to the tip. Pop out the yolks into a mixing bowl and put whites aside.
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Add the mayonnaise, vinegar, and mustard in a bowl. Use an electric mixer or mash the yolks using a fork until it is well combined, then taste (add Salt to taste). If the mixture is too thick for you, add a little more mayo. Then I push the mixture through a sieve or small-holed colander to get out any remaining yolk-lumps. This is optional if you actually like the lumps.
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You can just spoon into the whites, but filling egg whites using a pastry bag is so much prettier. In that case, fill a pastry bag with the egg yolk mixture, use star tip and pipe into the center of the whites.
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Garnish Deviled Eggs just before serving.