Cut chicken into bite-sized pieces and season with Italian seasoning and desired amount of salt and pepper.
Heat 2 tablespoons butter in a large pot over medium heat until melted. Add the chicken and cook until golden brown on all sides, about 5 minutes. Remove and set aside. Note: This step is not needed if using a cooked rotisserie chicken.
Add the chicken broth and rice to the pot and bring to a boil. Reduce heat to a simmer and cover tightly for 7 minutes.
Remove the lid and use a silicone spatula to give it a stir and loosen it from the bottom. Add the broccoli florets and replace the lid. Cook for more 8 minutes.
Stir the rice/broccoli and turn the heat off. Add the cooked chicken, soup, milk, sour cream, and half of the cheddar cheese.
Lightly grease a 9x13 casserole dish and pour in the rice mixture.
Top with remaining cheddar cheese.
Cover and bake for 20 minutes.
Add the Topping:
Melt the butter and crumble up the crackers. Combine to form the topping for the casserole.
Top the casserole with the crumbled cracker topping and bake uncovered for 10 additional minutes.
Let the casserole sit for 5 minutes prior to serving.