Place the eggs in a single layer in a saucepan and cover with cold water by an inch.

Bring to a boil over high heat, then cover, remove from the heat, and let stand for 10 minutes.

Transfer the eggs to a bowl of ice water and let cool for 10 minutes, then peel each egg under cool running water.

Pour the beet juice into a shallow bowl and submerge the peeled whole eggs for 2 hours, turning once halfway through, until the whites are a deep pink all the way around.

Remove the eggs from the beet juice and pat dry, then slice each in half lengthwise and scoop the yolks into a separate bowl.

Mash the yolks with the mayonnaise, mustard and vinegar until completely smooth with no lumps.

Season the yolk mixture with salt and pepper to taste.

Pipe or spoon the yolk mixture back into the pink egg white halves, mounding it slightly above the surface.

Garnish each with a small sprig of dill or a few snipped chives just before serving.