Preheat the oven to 375°F (190°C). Lightly grease a 9×13 inch casserole dish.
Heat olive oil in a large skillet over medium heat. Add the diced onion and cook for 3-4 minutes until softened.
Add the minced garlic and sliced carrots to the skillet. Cook for another 3 minutes, stirring occasionally.
Add the chicken pieces to the skillet. Sprinkle with dried oregano, dried thyme, salt, and black pepper. Cook for 5-7 minutes until the chicken is no longer pink on the outside.
Stir in the broccoli florets and cook for 2 minutes.
Pour in the chicken broth and bring the mixture to a simmer. Reduce heat and cook for 5 minutes until the vegetables are tender and the chicken is cooked through.
Remove the skillet from heat and stir in the Greek yogurt until well combined.
Add the cooked rice and half of the shredded mozzarella cheese to the mixture. Stir until evenly mixed.
Transfer the mixture to the prepared casserole dish. Sprinkle the remaining mozzarella cheese and the grated Parmesan cheese evenly over the top.
Bake in the preheated oven for 20 minutes or until the cheese is melted and bubbly.
Remove from the oven and let cool for 5 minutes before serving.