Put an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Butter a 13 by 9 by 2-inch baking dish. Set aside.
Sauce: In a 2-quart saucepan, melt the butter over medium heat. Add the flour and whisk until smooth. Gradually add the warm milk, whisking constantly to prevent lumps. Simmer over medium heat, whisking constantly, until the sauce is thick and smooth, about 8 minutes (do not allow the mixture to boil). Remove the pan from the heat and add the cheeses and nutmeg. Stir until the cheeses have melted and the sauce is smooth. Add the prosciutto, 1 1/2 teaspoons salt and 2 teaspoons pepper.
Filling: In a large skillet, heat the oil and butter over medium-high heat. Add the onion, season with salt and pepper, to taste, and cook, stirring frequently until soft, about 5 to 6 minutes. Add the garlic and cook until aromatic, about 30 seconds. Add the mushrooms, rosemary and thyme and season with salt and pepper, to taste. Cook, stirring occasionally, until the mushrooms are golden and the liquid has evaporated, about 8 to 10 minutes. Add 1 1/2 teaspoons salt and 2 teaspoons pepper.
Bring a large pot of salted water to a boil over high heat. Add the noodles and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Drain and set aside.
To assemble the lasagna: Ladle 1/2 cup of the sauce over the bottom of the prepared baking dish. Lay 3 pasta noodles on top. Spoon 1/2 of the mushroom mixture on top of the pasta. Ladle 1 cup of the sauce over the mushroom mixture and lay 3 more noodles on top. Repeat the layers using the remaining mushroom mixture, 1 cup of sauce and the remaining 3 noodles. Top with the remaining sauce and sprinkle with 1 cup smoked mozzarella cheese and 1/2 cup Parmesan. Drizzle the top with olive oil and bake until the top is golden and the filling is bubbling about 35 to 40 minutes. Remove from the oven and cool for 5 minutes. Cut the lasagna into wedges and serve.
This recipe has been updated and may differ from what was originally published or broadcast.