First, sauté mushrooms, spinach, and garlic in butter until tender. Then stir in flour and milk (and wine or broth) to form a creamy sauce, followed by cheeses until smooth and heated through. The filling becomes rich, flavor-packed, and ready to stuff.
Bring salted water to a boil and cook the large pasta shells until just al dente. You want them still slightly firm so they hold their shape when stuffed and baked. Drain carefully and set aside.
Lay the cooked shells open-side up in the baking dish. Spoon or pipe the ricotta-mushroom-spinach mixture into each shell generously. Pour any remaining sauce over the shells, and top with mozzarella, provolone, and Parmesan.
Cover with foil and bake in a preheated oven until the cheeses are melted and bubbly. Remove the foil for the final minutes so the top gets golden and slightly crisp. Let it rest for a few minutes before serving for the best texture.