Cook pasta as directed. Two minutes before the pasta is done, add broccoli to the pot. Set aside.

Heat olive oil over medium heat in a large skillet. Once the oil starts to shimmer, add the chicken: salt and pepper to taste. Cook the chicken for 4-6 minutes on each side. Once it's cooked through, transfer to a plate. Let it cool, then slice the chicken.

Melt butter in the same skillet over medium heat—no need to clean the skillet out. Whisk in flour and cook till the mixture is light brown.  

Add garlic, cook for 30 seconds. Whisk in milk and broth. Bring to a boil reduce to medium-low. Continue to whisk the sauce till it thickens, about 4-6 min. It's okay to stop whisking for a minute if you need to drain your pasta. 

Once the sauce has thickened, remove it from the heat and mix in the parmesan cheese. If the sauce is too thick, whisk in one teaspoon of milk at a time till it's the desired consistency. Salt and pepper to taste. 

 Add the pasta and broccoli to the skillet. Mix until everything is well coated. Top with the chicken.