Rinse and trim the green onions: finely chop 3 for the filling and thinly slice 2 for garnish. Mince 3 garlic cloves for the filling, smash 2 more for the broth, and grate 1 tablespoon ginger for the filling plus cut 4 thin slices for the broth.
In a large bowl, combine the ground chicken, chopped green onions, minced garlic, grated ginger, soy sauce, oyster sauce, Shaoxing wine, 2 teaspoons sesame oil, cornstarch, salt, 1/2 teaspoon white pepper, and egg white. Stir in one direction for about 2 minutes until sticky and paste-like, then chill for 20 minutes.
Line a rimmed baking sheet with parchment paper. Set out the filling, a teaspoon, and a small bowl of water, and keep the wonton wrappers covered with a damp kitchen towel.
Lay a wrapper like a diamond, add about 2 teaspoons filling, moisten the top edges with water, fold into a triangle, press out air, and seal. Dab water on one bottom corner, bring both corners together below the filling, and press to seal. Place on the sheet and cover with a damp towel; repeat with the remaining wrappers and filling.
Refrigerate covered if cooking within a couple of hours. To freeze, slide the sheet into the freezer for 1 to 2 hours until solid, then transfer in portions of 8 to labeled zip-top freezer bags.
Combine the chicken broth, ginger slices, smashed garlic, soy sauce, and 1/4 teaspoon white pepper in a medium pot. Boil, then reduce to low, cover partially, and simmer for 15 minutes. Taste and add salt if needed.
Bring 12 cups water to a gentle rolling boil in a large stockpot. Add about 24 wontons, stir once, and cook 3 to 4 minutes until they float and the wrappers are translucent. Lift out with a spider strainer; repeat with the second batch (add about 2 minutes if cooking from frozen).
Fish the ginger and garlic out of the broth with a slotted spoon. Simmer the bok choy for 1 to 2 minutes until crisp-tender and bright green, then turn off the heat and stir in 1 teaspoon sesame oil. Adjust salt or soy sauce to taste.
Divide the wontons among six warm bowls (about 8 each), tuck in 1 or 2 bok choy halves, ladle the hot broth over the top, and sprinkle with sliced green onions. Serve right away.