Place the dried porcini mushrooms in a heatproof bowl and cover with hot or boiling water; let soak 20–30 minutes, then reserve the soaking liquid and drain the mushrooms.
Make the marinara sauce or measure 3 cups of your favorite store-bought marinara and set aside.
Preheat the oven to 350°F (175°C).
Slice each zucchini lengthwise about 1/8 inch thick using a mandolin or sharp knife; season both sides lightly with kosher salt and let sit 10 minutes to draw out moisture.
Blot zucchini slices dry with paper towels, arrange in a single layer on a lined baking sheet, and bake 10 minutes to remove excess moisture; remove and set aside.
Heat the olive oil in a sauté pan over medium heat with a pinch of red pepper flakes and the garlic; cook 30–60 seconds until fragrant.
Add the diced onion (and any extra diced zucchini), season with salt, and sauté about 5 minutes until beginning to soften.
Add the diced crimini and shiitake mushrooms, dried oregano, dried thyme, and salt and pepper; cook until the mushrooms release their liquid and it mostly evaporates, about 15 minutes.
Add the rehydrated porcini mushrooms and about 1/4 cup of the reserved porcini soaking liquid; continue cooking until the liquid has evaporated and the mushroom mixture is nicely browned.
Spread about 1/4 cup marinara in the bottom of a 9x13 baking dish. Layer half of the baked zucchini slices over the sauce, top with half the mushroom mixture, spread half of the remaining marinara over the mushrooms, then sprinkle half of the Parmesan and half of the shredded mozzarella.
Repeat the layers once more with the remaining zucchini, mushroom mixture, marinara, Parmesan, and mozzarella.
Bake in the preheated oven until the cheese is browned and bubbly, about 40 minutes; tent with foil if the cheese browns too quickly, or briefly broil 1–2 minutes at the end if you want more browning.
Remove from the oven and let rest a few minutes before cutting. Serve with chopped basil and extra Parmesan if desired.