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 heat and cook chicken until golden and cooked through, 4–6 minutes per side depending on thickness; remove and 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.