Fill a large stockpot with water and bring to a boil. Stir in 1 teaspoon salt.
Add 8 ounces of uncooked Penne pasta to the boiling water and cook for 9 minutes total.
With 2–4 minutes remaining on the pasta timer, add 1 package (16 oz) frozen chopped broccoli to the pot. Continue boiling until the broccoli is cooked through.
Drain the pasta and broccoli in a colander and set aside.
While the pasta cooks, heat 2 tablespoons oil in a large skillet over medium-high heat. Add 1 lb chicken breast cut into cubes and 2–3 tablespoons Cajun seasoning; toss to coat.
Cook the chicken, stirring occasionally, until cooked through and no longer pink in the center, about 6–8 minutes. Transfer the chicken to a plate and keep warm.
Reduce the skillet heat to medium-low. Add 2 tablespoons butter; when melted, add 1 tablespoon minced garlic and cook, stirring, about 30 seconds until fragrant.
Sprinkle 2 tablespoons flour over the butter and garlic and whisk constantly for about 1 minute to cook the flour. Slowly whisk in 1/2 cup chicken broth until smooth, then whisk in 1 cup heavy whipping cream. Increase heat to medium and simmer, whisking, until the sauce begins to thicken and bubble, about 2–3 minutes.
Stir 1/2 cup shredded Parmesan cheese into the sauce until melted and smooth. Return the cooked chicken, drained pasta, and drained broccoli to the skillet. Toss everything together over low heat until evenly coated and heated through, about 1–2 minutes.
Serve immediately.