Melt the butter with the olive oil in a pot over medium-high heat. Add the onion and chicken and cook until chicken is no longer pink and onion softens, about 5 minutes.
Add the garlic, poultry seasoning, thyme, salt, pepper, garlic powder and onion powder. Stir.
Stir in the chicken bouillon.
Add the flour and cook for about 1 minute, stirring.
Stir in the chicken broth, mixing well.
Add the milk and stir well.
Add the rice, and bring to a low boil, stirring occasionally so the rice doesn’t stick to the bottom.
Cover and turn the heat down to low and cook for 15 minutes, or until rice is cooked, stirring occasionally so rice doesn’t stick.
Add the broccoli and stir in. Cover again and cook about 5 minutes, or until broccoli is tender.
Turn off the heat. Stir in half of the cheese, then top the casserole with the remaining cheese. Place the lid back on and let sit a few minutes until the cheese melts.
Serve garnished with green onions, chives, or parsley, if desired.