Preheat the oven to 375°F. Prepare an oven-safe Dutch oven or lidded oven-safe pot with at least a 3-quart capacity.
Melt the butter over medium heat. Cook the onion for about 4 minutes, until tender. Add the garlic and cook for approximately 1 minute.
Stir in the softened cream cheese, oregano, and basil until smooth. Add the broth, instant rice, and chopped broccoli. Bring to a boil, immediately turn off the heat, cover tightly, and leave undisturbed for 5 minutes.
Confirm that the rice is tender. Stir in the cooked chicken, sour cream, 1 cup cheddar, Parmesan, salt, and pepper.
Top with the remaining 1 cup cheddar and breadcrumbs. Bake uncovered for approximately 20 minutes, until lightly bubbling, melted and lightly browned on top, and 165°F in the center.