Prepare the Dried Mushrooms : Soak the dried mushrooms in boiling water for 20-30 minutes until softened. Strain the mushrooms, reserving the soaking liquid, and finely chop the rehydrated mushrooms. Set aside.
Cook the Mushroom Filling : Heat olive oil in a large skillet over medium heat. Add minced shallots and garlic, cooking until fragrant. Stir in fresh mushrooms and cook until browned and softened. Add the chopped rehydrated mushrooms, red wine, reserved soaking liquid, thyme, salt, and pepper. Simmer for 10-15 minutes, stirring occasionally, until the liquid has reduced. Set aside.
Prepare the Béchamel Sauce : In a saucepan, melt butter over medium heat. Whisk in the flour and cook for 1-2 minutes to form a roux. Gradually add milk, whisking constantly to avoid lumps. Add salt, pepper, and nutmeg. Simmer until thickened and smooth. Remove from heat.
Make the Ricotta Mixture : In a bowl, combine ricotta, egg, Pecorino Romano or Parmesan, parsley, salt, and pepper. Mix until well combined.
Assemble the Lasagna : Preheat the oven to 180°C (350°F). Spread a thin layer of béchamel on the bottom of a baking dish. Place a layer of lasagna noodles over the béchamel. Add a layer of mushroom filling, followed by dollops of ricotta mixture and béchamel Repeat layers until all ingredients are used, finishing with béchamel and grated mozzarella on top.
Bake the Lasagna : Cover the dish with foil and bake for 25 minutes. Remove the foil and bake for an additional 15-20 minutes, or until the top is golden and bubbling. Let the lasagna rest for 10 minutes before slicing.