1. Preheat the oven to 230°C fan.
2. Put the mushrooms and oyster mushrooms in a food processor in three or four batches, and finely chop each batch using the pulse function (or finely chop them by hand). Toss the chopped mushrooms with 3 tbsp oil and 1 tsp salt in a large bowl, and spread them out on a baking tray lined with baking paper. Roast them in the oven on the top rack for 30 minutes, until they are golden brown and well shrunken. Stir them three times during cooking. Remove them from the tray and set aside. Reduce the oven temperature to 200°C, still fan.
3. Meanwhile, put the dried mushrooms in a bowl with the chilies and the hot broth, and let them soak for 30 minutes. Strain the broth into another bowl and press as much liquid out of the mushrooms as possible, until you have 3½ dl (top up with water if necessary). Roughly chop the soaked mushrooms (it’s fine if some pieces are larger), and finely chop the chilies. Set the broth and mushrooms aside separately.
4. Finely chop the onions, garlic, and carrots using the pulse function in the food processor, or chop them by hand. Heat ½ dl oil over medium-high heat in a large sauté pan. Add the onion mixture and sauté for 8 minutes, stirring regularly, until the onions are translucent and golden. Finely chop the tomatoes using the pulse function in the food processor (or by hand), and add them to the pan along with the tomato paste, 1½ tsp salt, and 1¾ tsp pepper. Cook for another 7 minutes, stirring regularly. Add the soaked dried mushrooms and chilies along with the oven-roasted mushrooms, and let it all simmer for 9 minutes without stirring – the mushrooms should be slightly crispy and browned at the bottom. Stir in the mushroom broth and 8 dl water, bring it all to a boil, then reduce to medium heat. Cook the sauce for about 25 minutes, stirring regularly, until it has the consistency of classic meat sauce. Stir in 1 dl of the cream and let the sauce simmer for another 2 minutes. Remove from heat.
5. Mix the two types of cheese and the herbs together in a small bowl. Assemble the lasagna by spreading one-fifth of the sauce in the bottom of a round 28 cm baking dish (or a 20 x 30 cm square dish), and sprinkle one-fifth of the cheese mixture on top. Lay a layer of lasagna sheets over it, breaking them into pieces to fit the shape of the dish. Repeat the layers three more times in the same order, finishing with a final layer of sauce and cheese – that is, five layers of sauce and cheese in total, separated by four layers of pasta.
6. Drizzle 1 tbsp cream and 1 tbsp oil over, cover the dish with foil, and bake the lasagna for 15 minutes. Remove the foil, increase the temperature to 220°C fan, and bake the lasagna for another 12 minutes, turning the dish halfway. Set the oven to grill and grill the lasagna for an additional 2 minutes, until its edges are brown and crispy. Let it cool for about 5 minutes, then drizzle with the remaining tbsp of cream and oil. Sprinkle with the remaining parsley, and finish with a good sprinkle of pepper.