Pour the brine of the pickled beets into a small saucepan (reserve the beets). Add the apple cider vinegar and brown sugar. Bring to a low boil and stir until the sugar dissolves. Let cool completely.
Fill a large saucepan with water and bring to a boil. With a ladle, slowly lower the eggs into the water one by one. Reduce the heat to a low boil and cook for 10 minutes; drain. When the eggs are cool enough to handle, remove the shells.
Place the eggs in a small bowl or a jar. Pour the cooled beet brine mixture over the eggs. Cover and refrigerate for at least 6 hours and up to 24 hours. The longer the eggs are in the brine, the deeper the shade of pink they will be.
Slice each egg in half lengthwise. Remove the yolks and place them in a food processor. Arrange the whites cut-side up on a serving platter. To the food processor, add the mayo, mustard, salt, pepper and one of the pickled beet slices. Process until well blended and smooth. Taste for seasoning. If you want a deeper pink, add another beet to the mixture and process (reserve the remaining beets for another use).
Transfer the mixture to a resealable plastic bag. With scissors, snip off a bottom corner from the bag. Using the bag like a pastry bag, pipe the yolk mixture into the egg whites. Garnish with minced fresh chives. Cover loosely and chill until ready to serve.