Cook the pasta in boiling water according to package directions, adding in the fresh broccoli florets the last minute of cooking time; drain (do not rinse) then transfer to a large bowl and toss with 1-2 tablespoon oil or butter to prevent sticking; cover lightly with foil to keep warm.
Heat oil and butter in a large skillet over medium heat.
Season the chicken cubes with salt and pepper; add to the skillet and cook for 4-5 minutes or until the chicken is cooked and browned (adding in the garlic the last 3 minutes of cooking time).
Add in flour, pinch dried thyme and cayenne pepper (if using); stir with wooden spoon for 1 minute.
Add in table cream and grated parmesan cheese; bring to a simmer, stirring constantly and cook 1-2 minutes and until smooth and thickened.
Add in salt and pepper to taste.
Pour the sauce over the cooked pasta and broccoli; toss to coat.