Put the steaks in a ceramic or glass dish. Mix the soy sauce, lime zest and juice, garlic, chilli and sesame oil. Rub the steaks well with this and marinate for 30 minutes.
Drain the steaks (but reserve the marinade), then fry or griddle in a heated pan for 4 to 5 minutes each side, depending on the thickness of the steak and on your liking (if you want it well done, cook for an extra 1 to 2 minutes each side).
Transfer the steaks onto a plate, cover and rest them.
Slice the mushrooms and cut the carrot into 5 cm long thins strips. Heat the groundnut oil in a wok and stir-fry the vegetables for 3 to 4 minutes. Add the reserved marinade and let it bubble for a minute or so. Turn off the heat, mix in the Mirin and serve onto two plates.
Put the steaks on top of the veggies and scatter the toasted sesame seeds all over.