Heat the oven to its highest setting (or 230C fan). In four or five batches, finely chop the fresh mushrooms in a food processor on the pulse setting, then tip into a large bowl and toss with three tablespoons of oil and a teaspoon of salt. Spread out on a large, 40cm x 35cm oven tray lined with baking paper, and roast for 20 minutes in the top third of the oven, stirring once halfway, until the mushrooms are golden brown. Remove and turn down the oven to 220C (200C fan).
Meanwhile, put the dried mushrooms, chillies and hot stock in a large bowl and leave to soak for half an hour. Strain into a second bowl, squeezing as much liquid out of the mushrooms as possible; you should get about 340ml liquid in total, so top up with water, if need be. Very roughly chop the rehydrated chillies and mushrooms. Put the onion, garlic and carrot in a food processor and pulse until finely chopped.
Heat 60ml oil in a large saute pan on a medium-high heat, then fry the onion mix for eight minutes, stirring occasionally, until soft and golden. Pulse the tomatoes in the food processor until finely chopped, then add to the pan with the tomato puree and a teaspoon and a half each of salt and finely cracked black pepper. Cook for seven minutes, stirring occasionally, then add the rehydrated mushroom mix and the roast mushrooms, and cook for seven minutes more, resisting the urge to stir too much: you want them to go slightly crisp and brown on the bottom. Stir in the reserved mushroom liquid and 800ml water and, once it comes up to a simmer, reduce the heat to medium and cook for 18 minutes, stirring occasionally, until the mix reduces to a ragu consistency. Stir in 100ml of the cream, simmer for two minutes, then turn off the heat.
Mix both cheeses and both herbs in a small bowl. To assemble the lasagne, spread a fifth of the sauce in the base of a round, 28cm baking dish, top with a fifth of the cheese, followed by a layer of lasagne sheets, broken to fit as necessary. Repeat these layers three more times, then finish with a final layer of sauce and top with the last of the cheese.
Drizzle a tablespoon of cream and a tablespoon of oil over the top, cover with foil and bake for 15 minutes. Remove the foil, turn up the oven to 240C (220C fan)/465F/gas 9, and bake for 10 minutes more, turning the dish halfway, until the edges are brown and crisp. Leave to rest for five or so minutes, then drizzle over the remaining tablespoon each of cream and oil, sprinkle with the extra parsley and serve hot.
