Drain the mushrooms (discard the water), then roughly chop and put in a medium-sized bowl. Mix a third of the spring onions with the ginger and set aside. Heat the oven to 180C.
Put a 28cm ovenproof saute pan for which you have a lid on a medium-high heat. Add three tablespoons of oil, the onions, garlic, lime peel, the remaining spring onions and half a teaspoon of salt, and fry, stirring frequently, for eight to 10 minutes, until soft and golden brown. If the onions start browning too quickly, reduce the heat to medium. Transfer the onions to the mushroom bowl and stir to combine. You’ll assemble the gratin in the same pan, so don’t worry about cleaning it.
Put the jerusalem artichokes and potatoes in a bowl, season with half a teaspoon of salt and a good grind of pepper, then gently toss. Layer a third of the jerusalem artichoke and potato mix over the base of the pan, then top with a third of the onion mixture. Repeat with two more layers of each, finishing with the onions.
Pour over the stock and coconut milk, and cook on a medium-high heat, until the stock begins to simmer around the edges – about two to three minutes – then cover and cook for five minutes more. Take off the lid, then transfer the gratin to the oven for 15 minutes. Turn the oven to its grill setting, brush the top of the gratin with the remaining tablespoon of oil, and grill for five to six minutes, until crisp and golden brown on top.
Drizzle the lime juice over the top, then scatter over the coriander, the reserved spring onion and the ginger. Leave to rest for five minutes or so before serving.
