Cook 12 oz. egg-based pasta (such as fettuccine nests) in a large pot of boiling salted water, stirring occasionally, until al dente, about 1 minute less than package directions.
Meanwhile, heat 2 Tbsp. extra-virgin olive oil in a large skillet over high. Add 1 lb. crimini or mixed mushrooms (such as button, maitake, oyster, and/or shiitake), sliced ¼" thick or torn into bite-size pieces, and ¼ tsp. Kosher salt and toss to coat evenly. Spread out into an even layer and cook, undisturbed, until mushrooms are starting to release their liquid, 2–3 minutes (do not be alarmed; there will be a lot). Continue to cook, tossing occasionally, until moisture is almost completely evaporated and edges of mushrooms are starting to brown, 6–8 minutes.
Reduce heat to medium-high; add 5 garlic cloves, thinly sliced, and remaining 1 Tbsp. extra-virgin olive oil. Cook, tossing often, until garlic is fragrant and golden, 1–2 minutes. Pour in ¾ cup dry white wine, scraping up any browned bits stuck to bottom of pan with a sturdy spatula or flat-edged wooden spoon, and cook until wine is mostly evaporated (you should have about 1 Tbsp. liquid left in pan), about 4 minutes. Reduce heat to medium-low. Add 4 Tbsp. unsalted butter, cut into pieces, and 2 Tbsp. white miso and stir until melted and combined. Slowly pour in ½ cup heavy cream, scraping up any browned bits stuck to bottom of pan; simmer until thickened slightly, 2–3 minutes. Add 1 Tbsp. low-sodium soy sauce; season with freshly ground pepper. Taste and add more salt if needed.
Using tongs, transfer pasta to mushroom mixture and add 1 cup pasta cooking liquid. Increase heat to medium and bring to a simmer. Cook, tossing constantly and adding ½–1 cup more pasta cooking liquid to reach desired sauce consistency if needed, until pasta is al dente and liquid has thickened slightly, about 3 minutes.
Remove pan from heat. Add 1 oz. Parmesan, finely grated (about ½ cup) to pasta and toss to combine. Season generously with more pepper; taste and season with more salt if needed.
Divide pasta among shallow bowls and top with more Parmesan and pepper.