Boil the Eggs: Place the eggs in a pot of water, bring to a boil, and cook for 10 minutes. Remove from heat, drain, and place in cold water.
Prepare the Dye: In a bowl, mix soy sauce and balsamic vinegar.
Dye the Eggs: Peel the boiled eggs and place them in the soy sauce and balsamic vinegar mixture. Let them soak for at least 3 hours or overnight for a deeper color.
Prepare the Filling: Cut the avocado in half, remove the pit, and scoop out the flesh into a bowl. Add mayonnaise, lime juice, Dijon mustard, salt, and pepper. Mash until smooth.
Assemble the Eggs: Slice the dyed eggs in half lengthwise. Remove the yolks and add them to the avocado mixture. Mix until smooth.
Fill the Eggs: Spoon the avocado filling into the egg whites. Garnish with paprika if desired.