Make the mushroom powder: Use a pastry brush to dust off the mushrooms. (They should be dry and brittle; if they’re soft, in a skillet over low heat, dry the mushrooms for 5–6 minutes, then cool completely.)

Break the mushrooms into small pieces into a food processor or spice grinder, then grind to a fine powder. Add the salt and sugar and process; set aside. (The powder will keep in an airtight container at room temperature for up to 6 months.)

Make the curry paste: In a medium pot over medium heat, cook the oil, ginger, garlic, and onion, stirring occasionally, until softened, about 5 minutes. Stir in the curry powder, turmeric, salt, sugar, and white pepper. Turn the heat to medium-low and cook, stirring frequently, until pasty and darkened, about 9 minutes. Stir in the chiles if desired, and cook for 1 minute more. Transfer to a blender and purée; set aside. (In an airtight container, the paste will keep in the fridge for up to 6 months.) Clean the blender and pot and return the latter to the stove.

Make the “fish” balls: Fill the pot with 4 inches of water and bring to a boil. Add the potatoes, turn the heat to medium-high, and simmer until fork-tender, 15–20 minutes.

Meanwhile, in the blender, purée the tofu, miso, and 3 tablespoons of water until smooth and set aside.

Drain the potatoes, then return to the pot and use a potato masher to mash well. Add the tofu mixture, potato flour, sweet potato flour, wheat starch, salt, sugar, white pepper to taste, and 1 teaspoon of the mushroom powder and mix to combine. (Reserve the remaining mushroom powder for another use.) Roll the mixture into 1¼-inch balls and transfer to a baking sheet.

Fill a large pot with 6 inches of water and bring to a boil. Working in batches, boil the balls until they float, 5–6 minutes. Using a slotted spoon, transfer to a large bowl and toss with up to 1 tablespoon of the oil to prevent sticking.

Make the sauce: In a small pot, whisk together the cornstarch and 2 tablespoons of water, then whisk in the coconut milk, soy sauce, 2 tablespoons of the curry paste, and 1 cup of water. (Reserve the remaining curry paste for another use.) Bring to a simmer over medium heat and cook until the sauce is thickened slightly, 2–3 minutes. Remove from the heat and cover to keep warm.

To a wok or skillet over high heat, add the remaining 2 tablespoons of oil. When it’s hot and shimmering, working in batches, fry the balls, turning occasionally, until deep golden brown and crisp, 3–5 minutes. Transfer to a large bowl and tent with foil. When all of the balls are fried, add the sauce and gently toss to coat. Serve hot, on skewers if desired.