In a medium pot, mix 1 cup apple cider vinegar, 1 tablespoon sugar, 2 1/4 teaspoons salt, and 3 cups water. Add the beet and bring to a boil, then remove from heat and let cool slightly.
Pour into a large heat-safe jar or bowl (including beet). Add 6 hard-boiled eggs, stir gently, and refrigerate for at least 3 hours or overnight for a darker shade of pink (stirring occasionally).
Remove eggs from liquid. Halve eggs. Place whites on paper towel to remove excess pink liquid and then place on a platter.
Gently transfer yolks to a bowl and smash. Add ½ of the egg (yolk and white) from the remaining hard-boiled (but not dyed) egg. Add mayonnaise, mustard, salt, pepper and ½ of the dill sprigs to the bowl and mix well.
Spoon or pipe the filling into the egg whites.
Garnish with additional dill.