Preheat oven to 350°F.

Cook pasta al dente** in a large stockpot of generously-salted water, according to package instructions.  About 2-3 minutes before the pasta reaches al dente, add in the broccoli florets and let them cook alongside the pasta for the remainder of the cooking time.  Drain and set aside.

Meanwhile, as the pasta water is heating, melt butter in a (separate) medium saute pan over medium-high heat.  Add garlic and saute for 1 minute or until fragrant, stirring occasionally.  Whisk in the flour until combined and cook for an additional 1 minute, whisking occasionally.  Slowly whisk in vegetable or chicken stock until the mixture is smooth.  Then slowly whisk in the milk until it is combined.  Continue cooking for 1-2 minutes, or until the mixture comes to a simmer.  Then remove from heat, and stir in the cheddar, Parmesan, salt and pepper until the cheese sauce is smooth.  Remove from heat.

Once the pasta and broccoli are cooked, pour the cheese sauce on top of the pasta, add in the chicken, and toss until everything is evenly combined.

At this point, you can either serve the pasta stovetop-style as-is.  Or pour the pasta into a greased 9 x 13-inch baking dish, and sprinkle with extra cheddar cheese and Panko breadcrumbs.  Bake for 15-20 minutes, or until the top becomes slightly crispy and the breadcrumbs are slightly golden.  Remove and serve immediately.

*To cook the chicken breasts on the stove top, heat 1 tablespoon olive oil in a large saute pan.  Pound chicken to even thickness, and sprinkle both sides with a few generous pinches of salt and pepper.  Add to the pan and cook for 3-4 minutes per side, or until the chicken is cooked through and no longer pink inside.  (Cooking times will vary depending on the thickness of your chicken.)  Or, to bake the chicken breasts in the oven, follow these instructions.

**If making the baked version of this dish, I recommend ever-so-slightly undercooking your pasta, as it will cook a bit more while it bakes in the oven.