Halve any larger chestnut mushrooms. Zest the lime and set the zest to one side. Squeeze the juice into a large bowl. Peel and grate the ginger. Finely chop the chilli, flicking out most of the seeds. Add the ginger and chilli to the lime juice.
Add the tamari and 2 tsp caster sugar to the bowl. Pour in 1 tbsp olive oil. Whisk together well. Add the whole shiitake mushrooms and chestnut mushrooms to the bowl and mix well to coat in the marinade. Set to one side.
Peel and finely chop the onion and carrot. Heat a pan. When warm, drizzle in 1 tbsp olive oil. Add the onion and carrot. Cook for 5 mins, stirring occasionally, till softened.
Rinse the rice under cold water and tip it into the pan with the softened veg. Add 2 tsp China 5 spice, the lime zest and stir. Pour in 300ml boiling water. Add a pinch of salt. Cover, bring to the boil, then turn the heat down. Simmer for 25 mins till all the water has been absorbed. Take off the heat and set aside, lid on, to steam in the pan for 5 mins.
While the rice simmers, finely shred the cavolo nero. Tumble the marinated mushrooms onto a foil-lined grill pan or tray. When the rice has 10 mins left to cook, heat your grill to a medium high setting. Slide the tray under the grill for 5 mins.
Add the shredded cavolo nero to the rice while it's steaming and place the lid back on. Baste the mushrooms with any remaining marinade. Put under the grill for a further 5 mins.
The mushrooms should be cooked and the cavolo nero wilted. Heap the spiced veggie rice onto a couple of place. Spoon over the yakitori mushrooms and any cooked marinade.