Cook linguine according to package directions; reserve 1/2 cup pasta water and drain.
Season chicken with salt and pepper. Heat olive oil in a large skillet over medium-high; cook chicken until golden and cooked through, 4–6 minutes per side depending on thickness. Remove; keep warm.
Lower heat to medium, add garlic and cook 30 seconds until fragrant. Deglaze pan with chicken broth, scraping browned bits.
Stir in cream, Parmesan, and red pepper; simmer until slightly thickened, about 2–3 minutes.
Return chicken and any juices to the pan, add pasta and basil; toss, adding reserved pasta water a splash at a time to loosen sauce.
Adjust seasoning, top with extra Parmesan, and serve immediately.