Place a large Dutch oven over medium-high heat. Add the olive oil and 2 tablespoons of the butter to the pan. Once the butter melts and starts to foam, add the diced onions to the pan and saute until translucent, about 3 to 4 minutes. Add the garlic to the pan and cook until fragrant, about 30 seconds. Add the mushrooms, thyme, and marjoram and cook, stirring often, until they have released their liquid and caramelized, about 12 to 15 minutes. Add 1 tablespoon of the flour and stir to combine. Add the wine and cook until mostly reduced. Add the chopped tomatoes, tomato paste, stock, 1/4 cup of the heavy cream, 1 1/2 teaspoons of salt and 1/4 teaspoon black pepper and cook, stirring occasionally, until the liquid has reduced and the sauce is thick enough to coat the back of a spoon, about 10 to 15 minutes. Taste and adjust seasoning, if necessary. Add the parsley to the pan, stir to incorporate and reserve while you prepare the bechamel.
Preheat the oven to 350 degrees F.
In a large saucepan heat 6 tablespoons of the remaining butter until melted. Add the remaining 6 tablespoons of flour and whisk to combine. Cook, stirring for 2 to 3 minutes; do not allow to color. Slowly whisk in the milk and increase the heat to medium-high heat. Cook, whisking constantly, until mixture comes to a boil and thickens, 4 to 6 minutes. Season with the remaining 3/4 teaspoon salt, pinch of white pepper, and the nutmeg. Whisk in 1 cup of the Pecorino Romano, cover, and set aside.
Using the remaining tablespoon of butter, grease a 10 by 15-inch (3-inches deep) baking dish or lasagna pan. Spoon 1/4 of the mushroom sauce onto the bottom of the dish. Cover with a single layer of no-boil lasagna pasta. Top the pasta with another 1/4 of mushroom sauce, then 1/3 of the bechamel sauce. Repeat the layering with more pasta, mushroom sauce, and bechamel, until all of the ingredients are used up. The top layer should be bechamel sauce. Sprinkle the top with the mozzarella cheese and the remaining pecorino. Drizzle the remaining 1/2 cup of heavy cream over the top, cover tightly with aluminum foil, and bake the lasagna, undisturbed, for 1 hour. Uncover the lasagna and continue baking until the lasagna is puffed, golden brown on top and a paring knife inserted into the lasagna meets with no resistance, about 20 minutes longer. Let the lasagna rest for 15 minutes before serving. Garnish with chopped fresh herbs.