In a large pot of boiling salted water, cook pasta according to package directions. In the final 3 minutes of cooking, add broccoli to the pasta and cook the remaining time. Drain pasta.
In a large 12 inch skillet over medium-high heat, heat oil. Add chicken to skillet and begin to cook, flipping them over once, until they are cooked through, about 2 minutes per side. Remove them from the pan and set aside.
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 1/3 cup Parmesan until well combined, about 1 minute. If the mixture is too thick, add more milk as needed.
Stir in pasta, broccoli and chicken. Serve and sprinkle with additional Parmesan cheese.