Heat the oven to 375°F. Set a 9×13-inch baking dish nearby.
Combine the dried porcini mushrooms and the boiling water in a heatproof bowl and soak until softened, 20–30 minutes. Lift out the porcini, strain the soaking liquid through a fine-mesh sieve lined with a paper towel, reserve 0.5 cup, and finely chop the porcini.
Heat the extra-virgin olive oil in a large skillet over medium-high. Add the onion and cook until translucent, 5–6 minutes. Add the cremini mushrooms and shiitake mushrooms and cook, stirring occasionally, until they release their liquid and it mostly evaporates, 7–9 minutes. Stir in the garlic and fresh thyme leaves and cook 1 minute. Add the chopped porcini, the dry white wine, and the reserved porcini liquid; simmer until nearly dry, 3–5 minutes. Stir in 1 tsp of the kosher salt and 0.5 tsp of the black pepper, reserving 0.75 tsp salt and 0.25 tsp pepper for the sauce and layering; remove from heat.
Cover the dish tightly with foil (tent so it doesn’t touch the surface) and bake for 25 minutes. Uncover and bake until bubbling at the edges and golden on top, 15–20 minutes more; broil 1–2 minutes if needed for deeper color.
Let the lasagna rest 15 minutes to set, then sprinkle with fresh parsley and cut into squares to serve.