Make the Pickled Pink Eggs:
In a small pot, combine the sliced beet, water, vinegar, sugar, and salt. Bring to a simmer, stirring to dissolve the sugar.
Remove from heat and let the mixture cool slightly.
Pour the beet liquid into a glass jar or bowl and add the peeled hard-boiled eggs.
Cover and refrigerate for at least 6 hours, or overnight for a deeper pink color.

Prepare the Deviled Egg Filling:
Remove the pickled eggs from the beet liquid and pat them dry.
Cut the eggs in half lengthwise and carefully scoop out the yolks into a bowl.
Mash the yolks with a fork, then mix in mayonnaise, mustard, vinegar, salt, pepper, and paprika until smooth.

Fill and Garnish:
Spoon or pipe the yolk mixture back into the pink egg whites.
Sprinkle with smoked paprika and chopped chives for an elegant touch.