1. Preheat the oven to 450 degrees Fahrenheit/220 degrees Celsius fan. Line a large, rimmed baking sheet with parchment paper.
2. Put the button mushrooms and oyster mushrooms into the bowl of a food processor, in three or four batches, and pulse each batch until finely chopped (or finely chop everything by hand). Toss the chopped mushrooms in a large bowl with three tablespoons of the olive oil and one teaspoon salt and spread out on the prepared baking sheet. Roast for 30 minutes near the top of the oven, stirring three times throughout, until the mushrooms are golden brown; they will have reduced in volume significantly. Set aside. Decrease the oven temperature to 425 degrees Fahrenheit/200 degrees Celsius fan.
3. Meanwhile, combine all the dried mushrooms, the chiles, and hot stock in a large bowl and set aside to soak for 30 minutes. Strain the liquid into another bowl, squeezing as much liquid from the mushrooms as possible to get just under one and 1/2 cups/340 milliliters—if you have any less, top up with water. Very roughly chop the rehydrated mushrooms (you want some chunks) and finely chop the chiles. Set the stock and mushrooms aside separately.
4. Put the onion, garlic, and carrot into the food processor and pulse until finely chopped (or finely chop everything by hand). Heat 1/4 cup/60ml olive oil in a large sauté pan or pot on medium-high heat. Once hot, add the onion mixture and fry for eight minutes, stirring occasionally, until soft and golden. Pulse the tomatoes in the food processor until finely chopped (or finely chop by hand), then add to the pan along with the tomato paste, one and 1/2 teaspoons salt, and one and 3/4 teaspoons pepper. Cook for seven minutes, stirring occasionally. Add the rehydrated mushrooms and chiles and the roasted mushrooms and cook for nine minutes. Resist the urge to stir; you want the mushrooms to be slightly crisp and browned on the bottom. Stir in the water and reserved stock and, once simmering, decrease the heat to medium and cook for about 25 minutes, stirring occasionally, until you get the consistency of a ragù. Stir in seven tablespoons/105 milliliters of the cream and simmer for another two minutes, then remove from the heat.
5. Combine both cheeses, the basil, and 1/2 cup/10 grams parsley in a small bowl. Spread one-fifth of the sauce in the bottom of a round 12-inch/30 centimeter baking dish (or a nine- by 13-inch/23- by 33-centimeter rectangular dish), then top with a fifth of the cheese mixture, followed by a layer of lasagne sheets, broken to fit where necessary. Repeat these layers three more times in that order, and finish with a final layer of sauce and cheese; that’s five layers of sauce, five layers of cheese, and four layers of pasta.
6. Drizzle with one tablespoon cream and one tablespoon olive oil, then cover with aluminum foil and bake for 15 minutes. Remove the foil, increase the temperature to 450 degrees Fahrenheit/220 degrees Celsius fan, and bake for another 12 minutes, rotating the dish halfway through. Turn the oven to the broil setting and broil for a final two minutes, until the edges are brown and crisp. Set aside to cool for five minutes or so, then drizzle with the remaining one tablespoon cream and one tablespoon olive oil. Sprinkle with the one teaspoon parsley and finish with a good grind of pepper. Serve at once.