Preheat oven to 350Fº degrees. Cut the green bell peppers into quarters and cut out white membrane and core. Lay the peppers on a piece of parchment paper, in a sheet tray. Roast for 30-35 minutes, until slightly charred. Take peppers out of the oven and leave to cool.
Leave the oven on.
Make the dengaku sauce by mixing grated ginger, white miso paste, sake, mirin and soy sauce in a bowl, until uniformly blended. Set aside.
Cut the chicken breasts into sizes equal to the quartered bell peppers. Lay a piece of chicken on top of a pepper – you can use the same parchment paper. Grab the dengaku sauce and with a brush, apply a layer on top of each chicken piece. Top with jalapeno and shiitake mushrooms.
Put back in the oven and cook for 15-20 minutes, until chicken is cooked through. Serve hot.