Add the beets, water, vinegar, and salt to a pot over medium heat. Simmer for 20 to 25 minutes covered, until the beets are tender.

Let the beet mixture cool completely, then transfer it to a large bowl. Add the peeled eggs, making sure they're completely covered, then chill (stirring once or twice), for at least 2 hours. Chilling overnight will give a thicker pink color.

Remove the eggs and pat dry. Slice them in half lengthwise. Remove the yolk to a small bowl with a spoon and place the egg whites on a plate.

Mash the yolks with a fork and add the mayonnaise, mustard, vinegar, salt and pepper. Stir everything together until it's smooth.

Use a spoon (or a piping bag) to add a portion of the deviled egg mixture back into the hole of each egg white. Sprinkle sliced chives and paprika on top for garnish.