Melt the butter in a large pot or Dutch oven over medium-high heat and brown the chicken pieces until golden on all sides. Season with salt, pepper, dried oregano, and dried basil.
Pour in the chicken broth and rice. Bring to a boil, reduce heat, and simmer, covered, for 5 minutes.
Add the broccoli florets and cook, covered, for 10 minutes, or until there is no more liquid in the pot and the rice is fully cooked. Turn off the heat and let it stand for 10 mins. Preheat the oven to 350 F.
Mix in the cream of chicken soup, milk, sour cream, and half of the shredded cheddar.
Spoon the mixture into a greased 9x13-inch casserole dish, top with the remaining shredded cheddar cheese, and bake, covered with foil, for 15 mins.
Prepare the topping: melt the butter and mix with the crushed crackers.
After 15 minutes, remove the casserole from the oven and top it with the butter and cracker mixture. Bake, uncovered, for 10 minutes.