Heat olive oil in a large deep skillet or Dutch oven over medium-high heat.
Add the chicken pieces to the skillet and cook for 5-6 minutes until lightly browned but not fully cooked through. Remove chicken from skillet and set aside.
In the same skillet, add the diced 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 the rice with the oil and onion mixture.
Pour in the chicken broth and stir to combine.
Return the browned chicken pieces to the skillet, then add salt, black pepper, dried thyme, and paprika. Stir gently.
Bring the mixture to a boil, then reduce heat to low and cover the skillet with a tight-fitting lid.
Simmer for 15 minutes.
After 15 minutes, remove the lid and add the chopped broccoli florets evenly over the rice and chicken mixture. Do not stir.
Cover again and cook for an additional 10 minutes until the rice is tender and the broccoli is cooked but still bright green.
Remove the skillet from heat and sprinkle shredded cheddar, mozzarella, and Parmesan cheeses evenly over the top.
Cover the skillet and let sit for 5 minutes to allow the cheese to melt.
Gently fluff the casserole with a fork before serving.