01 - Bring a large pot of salted water to a boil. Add the pasta and cook according to package instructions until al dente. Add the broccoli florets to the boiling pasta water for the last 3 minutes of cooking. Drain pasta and broccoli, reserving 1/2 cup of pasta water.
02 - Heat olive oil and butter in a large skillet over medium-high heat. Add the chicken pieces, season with salt and pepper, and sauté until golden and cooked through, about 6–7 minutes. Remove chicken to a plate and set aside.
03 - In the same skillet, add garlic and cook for 30 seconds until fragrant. Pour in the heavy cream and milk, stirring to deglaze the pan.
04 - Lower the heat to medium. Add the Parmesan cheese and whisk until the sauce is smooth and slightly thickened, about 2–3 minutes. If the sauce is too thick, add reserved pasta water a little at a time.
05 - Return the chicken, pasta, and broccoli to the skillet. Toss everything together until evenly coated in the creamy sauce. Adjust seasoning with salt and pepper to taste.
06 - Serve immediately, garnished with chopped parsley and extra Parmesan if desired.