Cook pasta according to package; add broccoli in last 3 minutes. Reserve 1/4 cup pasta water, drain.
Season chicken with salt and pepper. Sauté in olive oil over medium-high heat until golden and cooked through, about 6–8 minutes. Remove and keep warm.
Lower heat, melt butter, sauté garlic 30 seconds. Add cream and simmer 2–3 minutes.
Stir in Parmesan until smooth; add nutmeg, salt, and pepper. Thin with reserved pasta water if needed.
Return chicken, toss with pasta and broccoli until coated and heated through. Serve with extra Parmesan.