Place lime juice, tomatoes, carrots, and sesame seeds in jar, cover, shake well, refrigerate.
Stir soy, oil, salt, pepper,and syrup/honey in a jar or bowl, cover, and refrigerate.
Bring 2 quarts of water to boil. Drop in sliced romaine. Push down with stirrer or Chinese strainer—15-20 seconds, till limp. Remove with strainer, run under cold water and dry with paper towels. Chop fine.
On attractive platter (black, if you have one), arrange the romaine in a circle.
Spoon apples in an inner circle.
Spoon apples in an inner circle.
Cover with wax paper, cover that with plastic wrap, refrigerate up to four hours.
When serving, uncover platter. Shake the jar with the last five ingredients and pour over the carrots and romaine. Decorate with carrot shreds and optional sesame seeds, and serve.