01 - Bring a large pot of salted water to a boil. Add pasta and cook according to package instructions until al dente. Add broccoli florets during the last 3 minutes of cooking. Drain pasta and broccoli together, reserving ½ cup of pasta water.
02 - Heat olive oil and butter in a large skillet over medium-high heat. Add chicken pieces, season with salt and pepper, and sauté until golden and cooked through, approximately 6-7 minutes. Transfer chicken to a plate.
03 - In the same skillet, add minced garlic and cook for 30 seconds until fragrant. Pour in heavy cream and milk, stirring to deglaze the pan.
04 - Reduce heat to medium. Add Parmesan cheese and whisk until the sauce is smooth and slightly thickened, approximately 2-3 minutes. Add reserved pasta water gradually if the sauce becomes too thick.
05 - Return chicken, pasta, and broccoli to the skillet. Toss all ingredients until evenly coated in creamy sauce. Adjust seasoning with salt and pepper as needed.
06 - Transfer to serving dishes immediately. Garnish with chopped parsley and extra Parmesan cheese if desired.