Boil the pasta in a large pot of salted water according to the package directions for al dente. During the last 2 minutes of cooking time, add the broccoli, cooking until the broccoli is bright green and just barely fork tender. Transfer the pasta and broccoli to a colander to drain.
Meanwhile, cut the chicken into ½ inch bite size pieces. Heat the oil in a large skillet over MEDIUM heat. Add the chicken and season it with all purpose seasoning and pepper. Sauté until the chicken is cooked through and just lightly golden (not browned). Transfer the chicken to a plate and set it aside.
Wipe the skillet clean, add the butter, and melt it over MEDIUM-LOW heat. Add the garlic and cook, stirring, for a minute or until fragrant. Whisk in the flour and cook, whisking, for a minute or two. Gradually add the milk (and/or half-and-half) and cook, stirring constantly with a wooden spoon until thickened. Reduce the heat to LOW and add the garlic powder, onion powder, and salt. Remove the pan from the heat and add ½ cup Parmesan and stir until the cheese has melted and the sauce is smooth. If the sauce is too thick, add a little additional milk or half-and-half to thin it, as needed.
Add the cooked pasta, broccoli, and chicken to the alfredo sauce in the skillet and toss to combine well. Top with additional Parmesan cheese before serving.