In a 4-quart saucepan, bring the chicken broth, 2 Tbs. of the soy sauce, and the sliced ginger to a boil over medium heat. Reduce the heat to low and let the ginger steep while you make the wontons.
Combine the pork, the remaining 1 Tbs. soy sauce, the grated ginger, scallion whites, rice wine, cornstarch, and sesame oil in a medium bowl and mix gently by hand.
Arrange the wonton wrappers on a clean work surface and put 1 tsp. of the pork mixture in the center of each.
Fill a small bowl with warm water. Dip your finger into the water and run it along the edges of a wonton wrapper. Fold the wrapper diagonally over the filling, pressing out any air, to make a triangle. Press the edges firmly together. Moisten the two points on the long side of the triangle and pinch them together over the filling to seal. (The finished wonton should look like a pope’s hat.) Repeat with the remaining wrappers.
Remove the ginger from the broth with a slotted spoon. Raise the heat to a lively simmer, add the wontons one at a time (so they don’t stick together), and cook until the pork is done, 5 to 8 minutes. (To check, pull one wonton out and cut it open to make sure the pork is cooked through.)
With the slotted spoon, divide the won tons among 4 soup bowls. Divide the lettuce among the bowls. Season the broth to taste with 1 or 2 drops of sesame oil and divide evenly among the soup bowls. Sprinkle with the scallion greens and serve.