To prepare the dried mushrooms and chiles, put the dried mushrooms in a medium bowl and the chopped chilis in a separate small bowl. Pour enough hot stock over the chiles to cover them and pour the rest over the dried mushrooms. Let both stand 30 minutes or until softened. When soft, set a strainer over another bowl or measuring cup and strain the mushrooms, squeezing out as much moisture as you can.
Chop the mushrooms, leaving a few of them coarsely chopped, and put the mushrooms back in the bowl you rehydrated them in. Strain the chiles so their liquid goes into the same bowl as the liquid from the mushrooms, squeezing them to get as much moisture out as possible. (You may want to wear latex gloves if your hands are sensitive, and so you don't get chile in your eyes later if you rub them.) You should have about 1 1/2 cups (340ml) of liquid. If not, top it up to that amount with water. Finely chop the chiles and add them to the bowl of chopped mushrooms. Set the liquid and chopped rehydrated mushrooms and chiles aside.
To prepare the mushroom ragù (the lasagna filling), preheat the oven to 425ºF (220ºC) and set a rack on the top third of the oven. Working in 3-4 batches, chop the fresh mushrooms in a food processor (or by hand) until they're finely chopped. (No need to clean the food processor as you'll be using it again shortly.) Spread the mushroom pieces on a large baking sheet and drizzle with 3 tablespoons olive oil and 1 teaspoon of salt. Mix them together with clean hands, then spread in an even layer. Roast in the oven, stirring the mushrooms three times during baking, until they are dark, well-reduced, and slightly crisp, about 35 minutes.
Finely chop the onion, carrot, and garlic in the food processor (or by hand.) Heat 4 tablespoons (60ml) olive oil in a large skillet (preferably one that's at least 14-inches/35cm wide, with high sides, or use a Dutch oven) over medium-high heat. Add the chopped onion/carrot/garlic mixture and cook, stirring frequently, until the vegetables are soft and starting to brown, 6 to 7 minutes. While the vegetables are cooking, finely chop the tomatoes in the food processor, or by hand.
Add the chopped fresh tomatoes, tomato paste, 1 1/2 teaspoons salt, and 1 3/4 teaspoons black pepper. Reduce the heat to medium and cook for about 5 minutes, stirring occasionally.
Add the rehydrated chopped mushroom and chiles, as well as the oven-roasted mushrooms, and continue to cook until the mixture starts to brown on the bottom, about 9 minutes. (The original recipe said not to stir them while cooking, which I didn't read. Oops. I stirred them a few times and they came out fine.)
Stir in the reserved stock from rehydrating the mushrooms and chiles, the water, and cook, stirring occasionally, until it's a thick stew, about 25 minutes. Add 1/2 cup (125ml) of cream and continue to cook for two minutes. Remove from heat and when cool enough to taste, and season with fish sauce or soy sauce, as well as additional salt if desired.
Prepare the herb and cheese mixture by mixing the grated Parmesan and Pecorino cheeses together in a medium bowl with the chopped parsley and basil.
To assemble and bake the lasagna, preheat the oven to 400ºF (200ºC.) Spread some olive oil in a 12x8-inch (30x20cm) rectangular baking dish with high sides, or a similar-sized baking dish. Mine has 2-inch/5cm sides and it didn't overflow. If that's a concern, place the baking dish on a baking sheet lined with foil before baking.
Spread one-fifth of the mushroom ragù in the bottom of the baking dish. Sprinkle one-fifth of the cheese and herb mixture over the top. Place a layer of the lasagna noodles over the cheese, breaking any noodles to fill in any gaps. Continue to assemble the lasagna, repeating those three steps in that order, so you end up with five layers of sauce and cheese, and four layers of pasta.
Drizzle 1 tablespoon of olive oil and 1 tablespoon of heavy cream over the top. Cover with foil and bake for 25 minutes. (The original recipe said to bake for 15 minutes but my noodles didn't quite get cooked thoroughly in that time, but they did continue to "cook" after I removed the lasagna from the oven and let it sit for a bit. But I think adding 10 minutes to the baking time at this point will do the trick. Next time I make it, I'll update that to be sure.) Remove the foil and increase the heat of the oven to 450ºF (230ºC.) Bake for another 12 minutes, rotating the lasagna midway during baking.
Finally, turn on the grill or broil setting and cook for 2 minutes until the top is browned. (Make sure your baking dish is sturdy enough to stand up to being under the broiler for 2 minutes. Mine is enameled cast iron and it worked fine but if you are unsure, skip this step. You can check the website for your particular piece of bakeware to find out their recommendations.)
Remove the lasagna from oven and drizzle it with 1 tablespoon of cream and and 1 tablespoon olive oil. Sprinkle with the remaining tablespoon of chopped parsley and freshly ground black pepper right before serving.