Peel and smash 3 cloves of garlic. Place them in a large pot with the chicken, water and ginger. Bring to a boil and skim any residue from the top of the broth. Reduce heat, cover and simmer slowly for about 2 hours. Strain broth and season to taste with salt. Discard chicken bones.
Skim off fat from the top of the broth (or refrigerate overnight and remove hardened fat).
While the broth is simmering, chop the remaining garlic clove. Place it in the bowl of a food processor with the shrimp, 1/2 tablespoon green onions, grated ginger, celery, chopped carrot, sherry, stir-fry sauce, cornstarch and sesame oil. Pulse the food processor until a moderately fine texture is achieved. The mixture should not be a smooth puree.
Lay the won ton wrappers on a clean counter. Brush the edges of each wrapper with a little beaten egg. Place a rounded teaspoon of filling in the center of each wrapper. Bring all 4 corners of the wrappers together above the filling and pinch together firmly. Place on a paper towel-lined plate, cover and chill until ready to use. This should make at least 30 won tons. Make ahead and freeze if desired.
To assemble the soup, bring the chicken broth to a boil in a large pot and gently add the won tons. Slowly simmer for 5 to 8 minutes or until the shrimp is cooked and the wrappers are tender. About 2 minutes after adding the won tons, add the julienned carrot, bok choy, shiitake mushrooms and snow peas.
Divide soup and won tons among 6 soup bowls. Garnish with the remaining 3 tablespoons sliced green onions and toasted sesame seeds.
NOTES: Low-sodium canned chicken broth can be substituted for the chicken and water, but be sure to simmer with the ginger and garlic for at least 30 to 45 minutes.
To toast seeds, heat in a dry skillet over medium heat until they start to brown. Stir occasionally. Be careful not to burn.Makes 6 to 8 Source: