Toss the chicken pieces with salt, black pepper, and Cajun seasoning. Allow the chicken to marinate for a few minutes.

Cook the penne in generously salted boiling water until al dente, following the package instructions.

Steam or blanch the broccoli florets until vibrant and just tender, then drain and set aside.

Heat the butter and olive oil in a large sauté pan over medium-high heat. Add the seasoned chicken and sear it, cooking approximately 5 minutes per side until golden brown and cooked through.

Remove the chicken and keep it warm. In the same pan, reduce the heat to medium and add the minced garlic. Sweat the garlic for about 30 to 60 seconds until aromatic but not browned.

Pour in the heavy cream and gradually add the milk, stirring constantly until the mixture reaches a gentle simmer.

Stir in the grated mozzarella or Monterey Jack until completely melted and smooth.

Add the cooked penne, sautéed chicken, and prepared broccoli to the pan, ensuring everything is well-coated and warmed through.

Taste and adjust seasoning, adding more Cajun spice or salt and pepper if necessary.