Cook pasta according to package directions. Reserve 1 cup of the cooking water, then drain.
Meanwhile, heat oil in a large skillet over medium heat. Add garlic and crushed red pepper; cook, stirring once, until fragrant, about 1 minute. Add kale, mushrooms, thyme and salt; cook, stirring occasionally, until the vegetables are soft, about 5 minutes.
Stir in the pasta and enough of the reserved water to coat; cook, stirring, until combined and hot, about 1 minute more. Serve topped with Parmesan, if desired.
