Sprinkle the chicken pieces with ¼ teaspoon salt and a little pepper. Heat 1 tablespoon of the butter in a large pasta pot over medium-high heat. Add the chicken and cook until browned and cooked through, 5 to 7 minutes. Remove and set aside.
Fill the pot with salted water and bring to a boil. Then add the pasta and broccoli and cook the pasta halfway, about 5 minutes. Drain the pasta and broccoli, reserving at least 1 cup of the pasta cooking liquid, and then return the pasta and broccoli to the pot.
Reduce the heat to low. Stir in the milk, flour, the remaining 1 tablespoon butter, mustard, Worcestershire sauce and 1 teaspoon salt. Simmer, stirring, until the sauce thickens, 3 to 5 minutes.
Turn off the heat and stir in the grated cheeses, the chicken and ¼ to ½ cup of the cooking liquid to thin the mixture if desired, making sure to thoroughly combine. Transfer to a bowl and serve.