Preheat the oven to 350°F. Cut chicken into bite-sized pieces and season with salt, pepper, and Italian seasoning.
Heat butter in a large pot over medium heat until melted. Add the chicken and cook until golden brown, about 5 minutes. Remove the chicken and set aside.
Add the chicken broth, olive oil, and rice to the pot. Bring to a boil, then reduce heat to a simmer. Cover tightly and cook for 6 minutes.
Add the broccoli and replace the cover. Cook for 9 more minutes, or until no liquid remains in the pot and the rice is cooked through. If not cooked through, replace lid and simmer 5 more minutes.
Keep covered and remove from heat. Let the rice stand for 10 minutes, do not stir. Any rice on the bottom of the pot will release.
Add the cooked chicken, cream of chicken soup, milk, sour cream, optional seasonings, and half of the cheddar cheese. Stir to combine.
Transfer to a lightly greased 9x13-inch casserole dish and top with remaining cheese. Cover and bake for 15 minutes.
Combine the melted butter and crushed crackers. Sprinkle over the casserole and bake uncovered for 10 more minutes. Serve.
Notes
Frozen broccoli (about 2 cups) can be used in place of fresh; thaw and pat dry first, and consider steaming for a softer texture. Cream of chicken soup can be swapped for cream of mushroom, cream of broccoli, cheddar cheese, or broccoli cheese soup. Shredding cheddar from a block rather than using pre-shredded cheese helps it melt more smoothly. This casserole can be assembled and refrigerated up to 2 days ahead of baking; let it sit out 30 minutes and add 15 minutes to the covered bake time, adding the cracker topping during the last 10 minutes. It can also be assembled and frozen unbaked; thaw 24 hours in the fridge before following the same make-ahead baking steps. Leftovers keep in the refrigerator up to 3 days or in the freezer up to 3 months.