Bring a large pot of salted water to a boil. Add pasta and cook according to package directions. In the last three minutes, add broccoli florets.
Season chicken pieces with salt, pepper, and garlic powder. Heat a large pan over medium heat and add 1 tablespoon of butter and olive oil. Cook chicken until fully cooked, about 5 minutes, then transfer to a plate.
In the same pan, add the remaining 3 tablespoons of butter, heavy cream, and Parmesan cheese. Warm over low heat, stirring until cheese melts and sauce thickens. Add nutmeg and season with salt.
Drain pasta and broccoli, reserving some pasta water. Add to the sauce along with the chicken. Toss gently to coat, adding reserved pasta water if the sauce is too thick.
Serve hot with extra grated Parmesan on top.
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Notes
Store leftovers in an airtight container for up to 3 days. Reheat gently with a splash of cream or milk.