In a small bowl, combine the porcini mushrooms and boiling water. Soak until softened, about 15 minutes. Using a fork, remove the rehydrated mushrooms and squeeze any excess liquid back into the bowl; reserve the soaking liquid and finely chop the mushrooms. Set both aside.

While the porcini are soaking, in a food processor, working in 2 batches, pulse the fresh mushrooms until finely chopped, 6 to 8 pulses; transfer to a medium bowl and set aside.

In a 12-inch skillet over medium, melt 4 tablespoons butter. Add the garlic and cook, stirring, until fragrant, 1 to 2 minutes. Increase to medium-high, then add the porcini and chopped fresh mushrooms, the rosemary sprig and ½ teaspoon salt. Cook, stirring occasionally, until the liquid released by the mushrooms has evaporated and the mushrooms are lightly browned, about 15 minutes.

Add the wine and soy sauce, scraping up any browned bits, and cook until the liquid has evaporated, about 2 minutes. Pour the reserved mushroom soaking liquid through a fine-mesh sieve into the skillet, then add the broth. Bring to a boil, then reduce to medium-low and simmer, uncovered and stirring occasionally, until the mushrooms are tender and the sauce is very thick, about 20 minutes. Remove and discard the rosemary sprig.

Meanwhile, in a large pot, bring 4 quarts water to a boil. Add the pasta and 1 tablespoon salt, then cook, stirring occasionally, until al dente. Reserve about 1 cup of the cooking water, then drain; return the pasta to the pot.

When the mushrooms are tender, add the sauce to the pasta in the pot, along with the remaining 2 tablespoons butter, half of the parsley and ½ cup of the reserved cooking water. Cook over medium, stirring and tossing, until the sauce coats the pasta, 1 to 2 minutes; add more cooking water 1 tablespoon at a time as needed if the mixture looks dry.

Off heat, taste and season with salt and pepper. Serve sprinkled with the remaining parsley and cheese.