Heat the oven to 350°F, then cut the chicken into bite-sized pieces and season it with salt/pepper and Italian seasoning. Small, evenly cut pieces brown quickly and finish cooking evenly in the casserole.
Melt the butter in a large pot over medium heat and cook the seasoned chicken for about 5 minutes, then move it to a plate. Look for light golden edges and avoid crowding the pot so the chicken browns rather than steams.
Pour the chicken broth and olive oil into the same pot, add the uncooked rice, and bring the liquid to a boil before lowering the heat to a simmer. Cover tightly and let the rice cook for 6 minutes so it begins absorbing the seasoned broth.
Scatter in the broccoli, cover the pot again, and simmer for 9 minutes. When the liquid pockets have disappeared, test rice from the center; if it is still firm, cover quickly and cook for 5 more minutes.
Remove the pot from the heat and leave it covered for 10 minutes without stirring. The trapped steam finishes tenderizing the rice and helps grains release from the bottom of the pot.
Stir in the cooked chicken, cream of chicken soup, milk, sour cream, optional seasonings, and half of the cheddar cheese. The mixture should look creamy and evenly coated, with tender rice that separates instead of clumping.
Spoon the mixture into a lightly greased 9 x 13-inch casserole dish, scatter the remaining cheddar over the top, cover, and bake for 15 minutes. The cheese should melt into a bubbly layer around the edges of the dish.
Mix the crushed crackers with the melted butter, sprinkle them over the casserole, and bake uncovered for 10 more minutes. Take it out when the crumbs smell buttery and toasted and the top is golden.