Preheat the oven to 350°F (175°C). Lightly grease a 9×13-inch casserole dish with cooking spray.
Steam the broccoli florets until tender but still bright green, about 4-5 minutes. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Add the diced onion and cook until softened, about 3-4 minutes.
Add the minced garlic to the saucepan and cook for an additional 1 minute until fragrant.
Whisk in the flour and cook for 1-2 minutes to form a roux, stirring constantly.
Gradually whisk in the whole milk, ensuring no lumps form. Continue to cook and stir until the sauce thickens, about 5-7 minutes.
Remove the saucepan from heat and stir in the sour cream, salt, black pepper, paprika, and dried thyme.
Add the shredded cheddar cheese and half of the mozzarella cheese to the sauce, stirring until melted and smooth.
In a large mixing bowl, combine the shredded rotisserie chicken, cooked rice, steamed broccoli, and cheese sauce. Mix until everything is evenly coated.
Transfer the mixture to the prepared casserole dish. Sprinkle the remaining mozzarella cheese and grated Parmesan cheese evenly over the top.
Bake in the preheated oven for 30 minutes, or until the casserole is bubbly and the cheese on top is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.