Season the chicken pieces with salt, pepper, and Cajun seasoning if you’re using it. Let them sit a couple of minutes to absorb the flavors.

Cook the pasta al dente according to package directions, then drain. Meanwhile, cook or steam the broccoli until tender but still bright green. Set both aside.

In a large pot or skillet, heat the butter and olive oil over medium-high heat. Add the seasoned chicken and sauté ~5 minutes on each side (or until cooked through). Remove chicken and set aside.

In the same pot, reduce heat to medium. Add garlic and cook until fragrant (30 seconds to 1 minute). Pour in heavy cream, and gradually add milk, stirring constantly to avoid overheating. Bring nearly to a simmer.

When the sauce is warm, stir in the mozzarella (or Monterey Jack) cheese until melted, smooth, and well blended into the sauce.

Add the cooked pasta, chicken pieces, and broccoli into the sauce. Stir to combine everything, coat well, and heat through. Taste and adjust seasoning (more Cajun, salt, or pepper as desired).

Serve immediately while the dish is hot and creamy. Garnish with extra cheese, freshly cracked black pepper, or chopped parsley if desired.