Remove the shiitake stems and cut the caps into fourths, sixths, or eighths, depending on size. By hand, break the oyster mushrooms into smaller pieces.

Heat a large skillet over high heat. Add 3 tablespoons of the olive oil. When the oil is hot, sprinkle in the mushrooms. Don't stir! Let them sizzle until they have caramelized on the bottom, about 3 minutes. If you toss them too soon, they will release their liquid and begin to steam. When the bottoms are caramelized, lower the heat to moderately low, stir, season with salt and pepper, and continue cooking until they are tender, about 15 minutes. Stir in the garlic, parsley, and thyme and cook for a minute or two to release their fragrance.

Meanwhile, bring a large pot of salted water to a boil over high heat. Add the pasta and cook until al dente. Drain, reserving 1 cup of the pasta water, and transfer the pasta to a large bowl. Add the mushrooms to the pasta. Return the mushroom skillet to high heat, add the reserved pasta water, and cook, stirring to scrape up any stuck-on bits, until the liquid is reduced by half. Still on the heat, whisk in the sherry vinegar, then slowly whisk in the remaining 2 tablespoons oil.

Add the contents of the skillet to the pasta and toss well. Add the spinach and toss again. Season with salt and pepper. With a vegetable peeler, shave the Parmesan over the top. Toss again gently and serve immediately.