BOIL: Bring a large stock pot of water to boil. Salt generously and cook the pasta 1 minute less of package directions; save ½ cup of pasta water. Drain and set pasta aside for now.
CHICKEN: Season the chicken with salt, pepper, and garlic powder. Heat a casserole pan (or saute pan) over medium-high heat. Add the oil and cook the chicken for roughly 6-12 minutes or until the chicken is cooked through to 165ºF. Remove to a plate; keep warm. Once cooled, you can slice or dice it.
SAUCE: Over medium heat, add the butter to the same pan. Allow it to melt, then saute the garlic and red pepper flakes for 30 seconds before adding the flour. Let the flour cook for 1 minute. Slowly pour in the chicken stock as you whisk to scrape up and deglaze any stuck on bits. Allow the sauce to simmer and thicken slightly; about 1-2 minutes. Then pour in the half and half and pesto. Stir to combine and allow it to regain a simmer. Add the Parmesan cheese, pasta, and steamed broccoli. If the sauce thickens too much at any point, use the pasta water to thin it back out. Taste and adjust with additional salt or more pasta water as desired.
SERVE: Add the chicken to the serving dish and serve family style, or serve pasta in bowls and top with chicken.