Preheat oven to 350°F. Cook chicken, season with salt, pepper, and Italian seasoning. Sauté chicken in butter until golden, then set aside.
In the same pot, add chicken broth, olive oil, and rice. Bring to a boil, then simmer covered for 6 minutes. Add broccoli, cover, and cook for 9 minutes until rice is tender.
Let rice stand covered for 10 minutes. Mix in cooked chicken, cream of chicken soup, milk, sour cream, seasonings, and half the cheese.
Transfer mixture to a greased 9x13-inch dish, top with remaining cheese, and bake covered for 15 minutes. Then, sprinkle crushed crackers mixed with melted butter on top and bake uncovered for 10 minutes.
Notes
For extra flavor, add a pinch of paprika or hot sauce before baking.