Heat olive oil in a large deep skillet or Dutch oven over medium-high heat.
Add the chicken pieces and cook for 5-6 minutes, stirring occasionally, until lightly browned but not fully cooked through. Remove chicken from the skillet and set aside.
In the same skillet, add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 30 seconds until fragrant.
Add the uncooked rice to the skillet and stir to coat it with the oil, onions, and garlic.
Pour in the chicken broth, then add the salt, black pepper, paprika, and dried thyme. Stir well to combine.
Return the chicken pieces to the skillet, spreading them evenly over the rice mixture.
Bring the mixture to a boil, then reduce heat to low and cover with a tight-fitting lid.
Simmer for 15 minutes without lifting the lid.
After 15 minutes, add the chopped broccoli florets on top of the rice and chicken, then cover and continue cooking for another 10 minutes until the rice is tender and the broccoli is cooked but still bright green.
Remove the skillet from heat and gently stir in the sour cream, shredded cheddar cheese, and shredded mozzarella cheese until the cheese is melted and the casserole is creamy.
Let the casserole sit covered for 5 minutes before serving to allow flavors to meld.