3 cups beet juice (from a jar of pickled beets or bottled beet juice)
3 tablespoons mayonnaise
2 tablespoons Dijon mustard
1 tablespoon fresh lemon juice
1/4 teaspoon salt
1/4 teaspoon freshly ground black pepper
finely chopped chives, for garnish
Place the eggs in a large saucepan and add enough cold water so that it covers the eggs by at least an inch. Bring the water to a boil and then turn off the heat. Let the eggs sit in the hot water for 10 minutes. Transfer the eggs with a slotted spoon to a bowl of cold water and let cool for 30 minutes.
While the eggs are cooling, prepare two large glass jars with tightly fitting lids. Pour 1 1/2 cups of the beet juice into each of the jars. When the eggs have cooled, drain them, gently crack the shells, and peel the eggs. Place the peeled eggs into the glass jars (4 per jar) and add enough cold water so that the eggs are completely submerged. Place the lids on the jars and refrigerate for at least 6 hours or overnight.
Once the eggs are dyed, remove them from the beet juice (you can either discard it or use it to dye more eggs). Place the eggs on a cutting board and carefully slice in half lengthwise with a sharp knife. Gently remove the yolks from the eggs and transfer to a small bowl.
Arrange the dyed egg halves on a platter. Add the mayonnaise, mustard, lemon juice, salt, and pepper to the bowl with the egg yolks, and mash with a from until smooth. Fill the dyed egg halves with the deviled egg mixture using a small spoon. Sprinkle the chopped chives over the top. Serve immediately or cover (without garnishing) for up to two hours before serving.