Bring a large pot of salted water to a boil. Add the pasta and boil according to the package directions for al dente pasta. Add the broccoli florets to the pot with the pasta for the last 2 minutes of cook time. Drain the pasta and broccoli in a colander.
Place the large pot that you cooked the pasta in back on the stove, set over medium heat. Add the butter and cook until it melts. Add in the garlic and cook for 30 seconds more before adding in the flour. Cook for 2 minutes, stirring it occasionally.
Whisk in the milk and dry mustard. Bring to a boil and boil for 2-3 minutes, stirring occasionally, until mixture thickens.
Remove from the heat and whisk in the cheeses, a little at a time, making sure the previous addition has melted before adding more. If the sauce is too thick, add in another splash of milk until the desired consistency is reached. combine.
Stir in the broccoli. Optionally, you can also stir in the chicken or serve it with the pasta.
Portion into individual bowls or plates and top with sliced or cubed chicken. Serve while still hot.
Notes
Leftovers should be stored in an airtight container in the fridge for up to 5 days.Reheat individual portions in the microwave. You may need to add a splash of milk when reheating the pasta to get a saucy consistency. You can also reheat on the stovetop, keeping the heat low and stirring frequently.For the cooked chicken, you can use leftovers you have on hand, rotisserie chicken, or cook some fresh using your desired method.