1 pound dry fusilli pasta
1 (14- to 16-ounce) package frozen broccoli florets
6 tablespoons unsalted butter, divided
1 pound boneless, skinless chicken breasts, cut into 1-inch pieces
Freshly ground black pepper
1/4 cup all-purpose flour
5 cups whole or 2% milk
1 1/2 cups finely grated Parmesan cheese (about 6 ounces)
Bring a large Dutch oven or heavy-duty pot of heavily salted water to a boil over medium-high heat. Add the pasta and cook until almost tender (you want some bite left in the pasta, check it 1 minute before the time called for on the package).
Turn off the heat, add the broccoli to the pot, stir to combine, and let sit for 1 minute (the broccoli will not be cooked all the way through). Drain well and set aside.
Melt 2 tablespoons of the butter in the same pot over medium-high heat. Add the chicken in an even layer, season generously with salt and pepper, and cook, stirring every few minutes, until the chicken is browned and just cooked through, about 5 minutes total. Remove to a plate or bowl. Do not wash out the pot.
Return the pot to medium heat and add the remaining 4 tablespoons of butter. When the butter is melted, add the flour, scrape up the brown bits at the bottom of the pan, and stir constantly until the flour starts to smell toasty and darkens slightly in color, about 1 minute. While constantly whisking, add the milk until it is all incorporated and the mixture is smooth. Bring just to a boil, whisking constantly. Add the cheese and whisk until smooth. Turn off the heat.
Add the chicken and any accumulated juices, pasta, and broccoli, and stir to combine. Taste and season with salt and pepper as needed.
Divide the mixture between 2 (8- to 9-inch) baking dishes or disposable foil trays (try to get ones that are around 2 inches deep), or transfer all of the mixture into a 9x13-inch baking dish or disposable foil tray. Let cool completely, then wrap tightly in aluminum foil and freeze for up to 2 months.
Bake frozen, still covered in the foil, on the middle rack of a preheated 350°F oven until heated through, about 1 1/2 hours. Alternatively, thaw in the refrigerator overnight, then bake for only 1 hour.