In the food processor, combine the onion, garlic, and carrot, and pulse until finely chopped or finely chop everything by hand.
In a large sauté pan or pot over medium-high heat, warm 4 tablespoons oil. Once hot, add the onion mixture and cook, stirring occasionally, until softened and golden, about 8 minutes. In the food processor, pulse the tomatoes until finely chopped or finely chop by hand and then add them to the pan along with the tomato paste, salt, and pepper. Cook, stirring occasionally, for 7 minutes. Add the rehydrated mushrooms and chiles and the roasted mushrooms and cook for 9 minutes, stirring only once during the cooking. Resist the urge to stir more frequently; you want the mushrooms to be slightly crisp and browned on the bottom.
Stir in the water and reserved stock and bring to a simmer. Reduce the heat to medium and cook, stirring occasionally, until you get the consistency of a ragù, about 25 minutes.
Stir in 7 tablespoons of the cream and simmer for 2 minutes more, then remove from the heat.
In a small bowl, combine both cheeses, the basil, and 1/2 cup parsley.
Spread 1/5 of the sauce in the bottom of a round 12-inch (30cm) baking dish or a 9-by-13-inch (23-by-33-cm) rectangular dish, then top with 1/5 of the cheese mixture, followed by a layer of lasagna sheets, broken to fit where necessary. Repeat these layers 3 more times in that order. Finish with a final layer of sauce and cheese; that’s 5 layers of sauce, 5 layers of cheese, and 4 layers of pasta.
Drizzle with 1 tablespoon cream and 1 tablespoon olive oil, then cover with aluminum foil and bake for 15 minutes. Remove the foil, increase the temperature to 450°F (220°C), and bake for 12 minutes more, rotating the dish halfway through.
Turn the oven to the broil setting and broil until the edges are brown and crisp, about 2 minutes.
Set aside to cool for 5 minutes or so, then drizzle with the remaining 1 tablespoon cream and 1 tablespoon oil. Sprinkle with the 1 teaspoon parsley and finish with a good grind of pepper. Serve immediately.