Heat the oil in a heavy-bottomed skillet over medium-high heat, and cook the diced chicken breasts until slightly brown, about 4 mins. Add the onion, celery, bell pepper, garlic, oregano, basil, parsley, salt, and pepper for 3 more minutes.
Mix in the rice, cream of chicken soup, and chicken broth. Bring to a boil, reduce the heat, and simmer, covered, for about 15 minutes or until the rice is almost fully cooked, stirring occasionally.
Add the broccoli florets and ½ cup of shredded cheddar. Cook for 5 more minutes or until the broccoli florets are tender.
Sprinkle the remaining ½ cup shredded cheddar on top and cover until melted.