Heat the oil or butter in a pan, throw in the garlic and ginger and cook for 30 seconds, then throw in the mushrooms and stir-fry for 2 minutes. Pour over the stock, tamari, star anise and cinnamon stick and bring to a boil. Reduce the heat and simmer for 10 minutes.
Throw the carrots into the soup to cook for a further 7 minutes while you prepare the eggs.
In a separate pan half-filled with boiling water, gently lower in the eggs one by one and boil for 5 minutes for runny yolks and 9 minutes for hard-boiled. Drain and hold the eggs under cold running water to stop them cooking any longer and to cool them enough to peel. Peel under cold running water.
Ladle the soup into two bowls, slice the eggs in half and place on top of the soup. Throw the chilli over the soup to serve.
When you come across the cinnamon stick or star anise, remove them: they are there to add flavour to the soup but not to be eaten.