Preheat the oven to 425 degrees F. Line a baking sheet with parchment.
Brush the mushroom caps with 1 tablespoon of the olive oil, season both sides with salt and pepper and put on the prepared baking sheet with the interior sides down. Bake until shriveled and the liquid evaporates, about 30 minutes. Set aside to cool.
Leave the oven on but lower the temperature to 375 degrees F.
Add the cremini mushrooms, garlic and shallots to a food processor and pulse until the mixture is coarsely ground. Heat the remaining 1 tablespoon oil in a 12-inch skillet over medium-high heat. Add the mushroom mixture and stir. Cook, stirring every couple minutes, until the mushrooms begin to caramelize, about 8 minutes. Season with salt and pepper. Deglaze the pan with the Marsala and add the thyme. Set aside to cool.
Once cool, whisk together the mushroom mixture, ricotta and Parmesan. Season with salt and pepper. Fill each mushroom cap evenly with the mixture.
Spray the bottom of a 9-by-14-inch baking dish with cooking spray; spray one side of a piece of aluminum foil with the spray.
Ladle a thin, even coating of the Lightened-Up Bechamel into the prepared baking dish. Top with 4 lasagna noodles, a third of the spinach and another layer of bechamel. Next, place 6 stuffed mushrooms evenly in 2 rows. Add a third of the sliced mozzarella followed by a layer of bechamel. Repeat with 4 more noodles, aother third of the spinach and then some bechamel. Place the remaining 6 stuffed mushrooms evenly in 2 rows and top with another third of the mozzarella and then some bechamel. Add the remaining noodles and spinach, then finally another layer of bechamel and the remaining mozzarella on top.
Bake immediately, as the more time this sits, the more the shape and structure of the lasagna is compromised. Cover with the sprayed aluminum foil piece and bake on the middle rack until golden brown and bubbly, about 35 minutes. If you want extra golden brown cheese, place the lasagna in the broiler for a couple of minutes. Let the lasagna rest for at least 30 minutes before slicing with a sharp knife and scooping with a spatula. Garnish each slice with basil.
Heat the oil in a medium saucepan over medium heat. Add the flour, whisking until smooth, and cook for about 1 1/2 minutes. Slowly add the milk while whisking constantly to prevent lumps. Bring to a low simmer and add the Parmesan, nutmeg and salt and pepper to taste. Set aside.