Prepare the porcini mushrooms by placing them in a bowl with 2 cups of hot water. Let soak for 20 minutes, then remove the mushrooms, squeezing out excess liquid. Strain the soaking liquid through a fine sieve or coffee filter to remove grit, and reserve for later use. Chop the rehydrated porcini finely.
Clean and prepare the fresh mushrooms by wiping them with a damp paper towel (avoid washing if possible). Slice them uniformly, about 1/4 inch thick. Keep different varieties separate as they may require different cooking times.
Heat 2 tablespoons olive oil in a large sauté pan over medium-high heat. Add onions and cook until translucent, about 5 minutes. Add garlic and cook for another minute until fragrant.
Add fresh mushrooms in batches to avoid overcrowding the pan, which would cause them to steam rather than brown. Cook each batch until golden and they’ve released their moisture, about 5-7 minutes. Season with salt, pepper, and thyme as you go.
Add the chopped porcini mushrooms and white wine to the pan, scraping up any browned bits from the bottom. Let the wine reduce almost completely, then add 1/2 cup of the reserved porcini soaking liquid. Simmer until the liquid has reduced by half. Remove from heat and set aside.
Make the béchamel sauce by melting butter in a saucepan over medium heat. Add flour and whisk constantly for 2 minutes to create a roux. Gradually add warm milk, whisking continuously to prevent lumps. Cook for 5-7 minutes until thickened enough to coat the back of a spoon. Season with salt, white pepper, and nutmeg.
Preheat your oven to 375°F (190°C) and lightly oil a 9×13 inch baking dish.
Assemble the lasagne by spreading 1/2 cup of béchamel on the bottom of the dish. Place a layer of lasagne sheets on top, then spread with a third of the mushroom mixture, followed by a quarter of the remaining béchamel and a sprinkle of Parmesan.
Repeat the layering process twice more, finishing with a final layer of pasta, the remaining béchamel, and a generous topping of Parmesan cheese.
Cover with foil and bake for 40 minutes, then remove the foil and bake for an additional 15-20 minutes until the top is golden brown and bubbling.
Let rest for at least 15 minutes before cutting and serving to allow the layers to set properly.