To make picked eggs, in a large mason jar, mix together beet juice, apple cider vinegar and brown sugar. Add peeled eggs ensuring they are completely covered. Seal jar and refrigerate for 12 hours.
Drain eggs and cut in half lengthwise. Remove egg yolk and place in bowl. Add remaining ingredients except chives to egg yolks and mix well until creamy. Fill each egg white with egg mixture and top with chives.