Chop the mushrooms into rough 1cm slices. Scrub the carrot and use a peeler to create ribbons out of it Peel the garlic cloves and crush or finely chop them. Peel the shallots and finely slice them. Thinly slice the chilli on an angle. Separate the coriander leaves and stalks (save the leaves for later in the recipe), and thinly slice the stalks.
Fill a pan with boiling water and sprinkle in a little salt. Return to the boil and add the noodles. Simmer for 3 mins, then drain well. Rinse the noodles with plenty of cold water to cool down, then drain again and set aside.
Pour 2 tbsp oil into a large frying pan or wok and bring to a medium-high heat. Slide in the mushrooms and stir fry for 4-5 mins, till golden. Spoon in 1 tbsp of the brown rice miso, the sliced shallots, garlic, chilli, coriander stalks, carrot and 1 tbsp tamari. Fry for a further 2-3 mins, stirring frequently, till well combined.
Tip the noodles into the mushrooms and fold through the veg. Reduce to a low heat.
Pour 1 tbsp oil into a frying pan and bring to a medium-high heat. Crack in the eggs and fry for 3-4 mins, till crisp around the edges.
Pile the miso mushroom noodles onto a couple of warm plates and top with the fried eggs and reserved coriander leaves. Serve with the rest of the tamari on the side.