Heat the oil in a pan over medium heat.
Add the shiitake mushrooms and leek or spring onions, along with the soy sauce and Mirin, if used.
Cover and cook for about 5 minutes, until the mushrooms are tender.
Add the rice vinegar and cook for another minute, letting the mushrooms lightly caramelise.
Stir in the sesame oil, then remove from the pan and set aside.
In the same pan, cook the eggs to your liking - fried, scrambled or poached (in a separate pan of simmering water).
Spoon the mushrooms onto toast if using, then top with the eggs.
Finish with a sprinkle of sesame seeds and serve immediately.