Place the sliced ginger, unsliced bunch of scallions, ½ cup soy sauce, ¼ cup vinegar, garlic, and potatoes in a large pot with 6 cups water.
Simmer until the potatoes are cooked, about 8 minutes. Remove the potatoes and cut them into quarters.
Add the chicken to the pot. Simmer for 10 minutes or until the chicken is cooked through. Let rest for 5 minutes before slicing.
Meanwhile, bring a large pot of salted water to a boil. Add the green beans and cook until tender, 3 to 5 minutes; drain.
Place the sliced scallions, chopped ginger, and remaining soy sauce and vinegar in a bowl. Whisk in the honey, sesame oil, cayenne, and olive oil.
Add the green beans and potatoes to the dressing and toss. Divide among plates and serve with the chicken.