Preheat the oven to 375°F (190°C).
Heat olive oil in a large oven-safe skillet or casserole dish over medium heat.
Add the chicken pieces and cook until lightly browned but not fully cooked through, about 5 minutes. Remove chicken and set aside.
In the same skillet, add the diced onion and cook until softened, about 3 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Stir in the uncooked rice and cook for 1-2 minutes to lightly toast the rice.
Pour in the chicken broth, then stir in the sour cream, dried thyme, paprika, salt, and black pepper.
Return the chicken pieces to the skillet and stir to combine.
Arrange the trimmed green beans evenly over the top of the mixture.
Cover the skillet or casserole dish tightly with aluminum foil.
Bake in the preheated oven for 40 minutes.
Remove the foil and sprinkle the shredded cheddar and mozzarella cheeses evenly over the casserole.
Return to the oven uncovered and bake for an additional 10 minutes, or until the cheese is melted and bubbly and the rice is tender.
Remove from oven and let rest for 5 minutes before serving.