Thinly slice 2 inches of the ginger and add it to a large pot with 6 cups water, 1 bunch of the scallions, 1/2 cup of the soy sauce, 1/4 cup of the vinegar, the garlic, and potatoes. 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 it 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. Thinly slice the remaining scallions and finely chop the remaining ginger. Place them in a bowl with the remaining soy sauce and vinegar. Whisk in the honey, sesame oil, cayenne, and olive oil. Add the green beans and potatoes and toss. Divide among individual plates and serve with the chicken.