Place chicken stock in a large saucepan or Dutch oven over medium-high heat. Cut white parts of green onions into 1-inch pieces and add to broth. Add garlic clove. Thinly slice 1 inch ginger and add to stock. Add 1 tablespoon soy sauce, vinegar, brown sugar and 1/2 teaspoon salt. Bring mixture to a boil, stirring occasionally.

Place pork in a bowl. Finely slice remaining green parts of green onions. Set aside 3 tablespoons for garnish. Add remaining onions to the bowl; grate in remaining 2-inch piece of ginger. Add remaining soy sauce, remaining salt, sesame oil, and cornstarch to the bowl; mix until well combined.

Form mixture into 1-inch balls and add to broth. Alternatively, use a small spoon to drop heaping teaspoons full of meat into broth. Reduce heat to a simmer; cook 5 minutes.

Add wonton wrappers, bok choy, and additional soy sauce to taste. Cook for 8 minutes, stirring occasionally.