In a large pot away from the heat mix the shoyu, water, 1¼ cups rice vinegar, honey, garlic, bay leaf and black pepper; stir until the honey is dissolved.
Place the chicken into the large pot with marinade mix and bring to a boil over high heat. Reduce the heat to medium and simmer until chicken thighs are completely cooked, about 20 minutes. Remove the chicken from the marinade in the pot.
Reduce the marinade over high heat until 2 cups remain. Remove the bay leaves and discard. Set marinade reduction aside.
To make the miso-adobo butter, combine the marinade reduction, miso paste, 2 tbsp. rice vinegar, brown sugar, cornstarch and soft poached egg in a blender. Blend on high until smooth. While blender is running, drizzle in the hot melted butter in a slow steady stream until all of the butter has been added and the sauce is homogenous. Season to taste.
Place the chicken onto a platter and pour sauce over top. Garnish with green onions and romaine lettuce. Enjoy with your favorite winter veggies, rice and a glass of red wine.