Preheat the oven to 375°F (190°C).
Steam the broccoli florets until just tender, about 4-5 minutes, then drain and set aside.
In a large skillet over medium heat, melt the butter. Add the chopped onion and cook until translucent, about 4 minutes.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Sprinkle the flour over the onion and garlic mixture and stir constantly for 2 minutes to form a roux.
Gradually whisk in the milk and chicken broth, continuing to whisk until the sauce thickens, about 5-7 minutes.
Stir in salt, black pepper, dried thyme, and paprika into the sauce.
Remove the sauce from heat and stir in 1 cup of shredded cheddar cheese until melted and smooth.
In a large mixing bowl, combine the cooked chicken, steamed broccoli, cooked rice, and cheese sauce. Mix well.
Transfer the mixture into a greased 9×13 inch casserole dish.
Sprinkle the remaining 1/2 cup shredded cheddar cheese, grated Parmesan cheese, and breadcrumbs evenly over the top.
Bake in the preheated oven for 25-30 minutes, or until the top is golden brown and bubbly.
Remove from the oven and let the casserole rest for 5 minutes before serving.