Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch casserole dish.
Heat the olive oil in a large skillet over medium heat. Add the chopped onion and cook until translucent, about 4 minutes.
Add the minced garlic and cook for an additional 1 minute until fragrant.
Add the chicken pieces to the skillet and cook until they are no longer pink in the center, about 6-8 minutes. Season with salt, black pepper, paprika, and dried thyme while cooking. Remove from heat and set aside.
Steam the broccoli florets until just tender, about 4-5 minutes. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the chicken broth and milk, continuing to whisk until the sauce thickens and is smooth, about 5-7 minutes.
Remove the sauce from heat and stir in the sour cream until fully incorporated.
In a large mixing bowl, combine the cooked jasmine rice, cooked chicken, steamed broccoli, and the creamy sauce. Mix well.
Fold in 1/2 cup of the shredded cheddar cheese and 1/2 cup of the shredded mozzarella cheese.
Transfer the mixture to the prepared casserole dish and spread evenly.
Sprinkle the remaining cheddar and mozzarella cheese evenly over the top.
Bake in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese is melted and golden brown.
Remove from the oven and let it rest for 5 minutes before serving.