Place the quail eggs into a 2 quart saucepan with 3 cups of the water and slowly bring it to boil, covered. Remove the lid and continue to boil for 2 minutes. Take out and place under the cold running tap water for 10 minutes and shell for later use.
Discard the boiling water and put the remaining 3 cups of water into the saucepan. Add the star anises and salt to simmer for 20 minutes then mix in the dark soy, chicken essence and sugar to cook till dissolved. Turn off the heat and leave aside to cool for 20 minutes. Soak the eggs in the sauce for an hour. Remove with a slotted spoon, drain and coat with the sesame oil.
Line the wok with tin foil and sprinkle the tea leaves and brown sugar on top. Put in a rack and warm the wok with moderate heat until smoke rises. Arrange the eggs on the rack and cover to smoke for 15 minutes. Remove. Place the quail eggs on a plate lined with the shredded lettuce and carrots. Serve.