Heat the oil in a frying pan over medium heat. Add the onion and a little salt and cook, stirring, for 10 minutes then add the mushrooms and cook for a further 5 minutes. Set aside.
Cut the chicken breasts in half to get 4 thin fillets of an even thickness. In a shallow dish, combine the 2 tablespoons of miso with the mirin and vinegar. Add the chicken, stirring to coat well. Leave to marinate for 10 minutes.
In a small bowl, combine the mayonnaise with the 2 teaspoons miso paste. Heat a barbecue, grill pan or frying pan to medium. Cook the chicken pieces for 4 minutes then turn, top each with a piece of cheese, and cook for a further 3 minutes or until cooked through and the cheese has melted.
Spread some miso mayonnaise onto the warmed bun bases. Top with lettuce, chicken, the mushroom mix and the bun tops. Serve with pickles.