In a large skillet, heat oil over medium-high heat; add sausage, onion, and garlic, breaking up sausage with a spoon. Cook until onions are golden, about 7 minutes. Add mushrooms, thyme, and chiles; cook, stirring frequently, until mushrooms release their liquid and it evaporates, 10 to 12 minutes. Remove from heat; season with salt and pepper.
Meanwhile, bring a large pot of water to a boil; add salt and pappardelle. Cook until al dente, 3 to 5 minutes. Reserve a 1/2 cup cooking liquid; drain pasta. Return to pot.
Add butter to mushroom mixture along with some of the cooking liquid to loosen, if necessary. Return mushroom mixture to pot; toss to combine. Transfer to a large serving platter. Garnish with parsley and cheese; serve immediately.