Stir water, soy sauce, sake, and mirin together in a large pot. Lay chicken into the liquid with the skin-side down. Place pot over medium heat, cover pot with lid, and bring liquid to a boil; cook at a boil, stirring occasionally, until chicken is no longer pink, about 5 minutes.
Push chicken to the sides of the pot to make room for the cabbage; put cabbage into the liquid with the cut-side down. Replace lid on pot, bring liquid again to a boil, and cook until cabbage softens, about 5 minutes more.
Dot top of mixture with butter cubes. Replace lid and cook just until butter melts, 1 to 2 minutes. Transfer everything to a serving dish and garnish with green onion.