Cook broccoli in boiling water for 5 minutes, then drain well. Cut into small florets.

Cook pasta al dente according to package instructions, then drain.

In a large skillet, heat olive oil and butter over medium-high heat until hot but not smoking. Season chicken breast generously with salt and pepper on both sides, then sauté in the skillet for about 5 minutes on each side or until cooked through. Remove the chicken from the skillet, cover with a lid, and let it rest off heat until no longer pink in the center. Cut chicken into bite-size pieces.

Add minced garlic to the same skillet and cook in the remaining chicken juices and oil over medium heat for about 30 seconds until fragrant, being careful not to burn. Add the cooked and sliced broccoli to the skillet and mix with garlic. Remove from the skillet.

To make the creamy sauce, heat up 1 cup of heavy cream in the same skillet until it boils. Immediately add the cheese and stir until melted and creamy. Reduce heat to medium-low, then slowly add milk in small portions, constantly stirring and simmering until a creamy consistency is reached.

Add the cooked pasta to the creamy sauce, then add the broccoli. Taste and season with salt and pepper as needed. Top with the sliced, cooked chicken.