Prepare the pasta according to the package directions. In the final 3 minutes of cooking, add broccoli to the pasta and cook the remaining time. Drain pasta.
Cook the chicken:
In a 12-inch skillet, heat the oil over medium-high heat. Add the chicken to the skillet and cook for 2 minutes before flipping and cooking an additional 2 minutes. Remove the chicken pieces from the pan and set aside.
Make the sauce:
Melt butter in the skillet over medium heat. Whisk in flour until lightly browned, about 1 minute.
Gradually whisk in chicken broth and milk, and cook, whisking constantly until incorporated, about 1-2 minutes. Stir in heavy cream, garlic powder, and Italian seasoning until slightly thickened, about 1-2 minutes.
Add in ⅓ cup Parmesan until well combined, about 1 minute. If the mixture is too thick, add more milk as needed.
Combine:
Stir in the pasta, broccoli, and chicken. Serve and sprinkle with additional Parmesan cheese.
To pack in a thermos:
Preheat the thermos container with boiling water. This step is optional, but it helps.
Heat the chicken Alfredo in the microwave or stovetop to a piping-hot temperature.
Drain the water from the thermos.
Fill the thermos with the very hot chicken and broccoli Alfredo.
Close the thermos container and pack it inside a lunch bag with your choice of snacks.