Heat the 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 the 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 with the oil, onion, and garlic mixture.
Pour in the chicken broth and stir to combine.
Return the browned chicken pieces to the skillet and add the chopped broccoli florets.
Season with salt, black pepper, dried thyme, and smoked paprika. Stir gently to combine all ingredients.
Bring the mixture to a boil, then reduce heat to low. Cover the skillet with a tight-fitting lid and simmer for 20 minutes, or until the rice is tender and the chicken is cooked through.
Remove the skillet from heat. 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.