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In a large pot of boiling salted water, cook pasta, stirring occasionally, until al dente according to package directions. Reserve ½ cup pasta water, then drain.
Meanwhile, in a large, high-sided skillet over medium heat, heat oil. Add garlic, thyme, and red pepper and cook, stirring, until fragrant, about 1 minute. Add tomatoes, then slowly pour in cream and broth. Bring to a simmer. Add Parmesan and cook, stirring, until cheese is melted and sauce is starting to thicken, about 5 minutes; season with salt and black pepper.
Add pasta and ¼ cup reserved pasta water and stir to coat. Cook, stirring frequently and adding more pasta water as needed, until sauce is glossy and fully coats pasta, about 2 minutes more.
Divide pasta among bowls. Top with Parmesan and basil.