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.