Clean and slice mushrooms to about 1/4-inch thickness.
Heat 2 tablespoons of the olive oil and 2 tablespoons butter in a large skillet. Once the butter has melted, add half the mushrooms, sprinkle with 1/2 tsp salt, and cook over medium heat until the mushrooms have released their juices and are cooked through. Transfer to a bowl, and repeat with the remaining mushrooms. Set aside.
Make a roux. Melt remaining (1 stick) butter In a medium-sized saucepan over medium heat. Once melted, add the flour. Stirring constantly with a whisk, cook the roux until the flour-butter mixture starts to bubble. While whisking, cook an extra minute.
While whisking constantly, pour in the milk in three additions, starting with about a cup, then another cup, then the remaining 2 cups.
Once you’ve added all the milk, add 1 1/2 teaspoons of salt, the pepper, and the nutmeg.
Whisk constantly over medium heat until the cream sauce thickens and bubbles.
Preheat oven to 375°F.
If you’re using home-scratched pasta sheets, boil about 4 quarts of water containing the remaining 1/2 teaspoon of salt and about 1 tablespoon of olive oil to prevent the sheets from sticking together. Boil the sheets for about 2 minutes. If you’re using store-bought pasta, follow package directions.
Assemble lasagna. Spread some of the cream sauce on the bottom of the pan. Add a layer of pasta, then more sauce, then about 1/3 of the mushrooms and 1/4 cup of Parmesan cheese. Follow this with two more layers of pasta, sauce, mushrooms, and cheese. Top with a final layer of pasta, the remaining sauce, and the remaining cheese.
Place in the oven and bake for about 45 minutes, or until the sauce is bubbly all the way through and the top is lightly browned.
Once baked through, remove from the oven and let settle for about 15 minutes before serving.