Preheat the oven to 375°F (190°C).
In a large ovenproof baking dish (approximately 9×13 inches), combine the rinsed rice, chicken broth, water, chopped onion, minced garlic, frozen peas and carrots, sour cream, dried thyme, salt, and black pepper. Stir well to combine.
Drizzle 1 tablespoon of olive oil over the chicken breasts and rub evenly. Sprinkle paprika over both sides of the chicken breasts.
Arrange the chicken breasts on top of the rice mixture in the baking dish.
Drizzle the remaining 1 tablespoon of olive oil over the chicken breasts.
Cover the baking dish tightly with aluminum foil and place it in the preheated oven.
Bake for 40 minutes covered. After 40 minutes, remove the foil, sprinkle shredded cheddar cheese evenly over the chicken and rice.
Return the dish to the oven uncovered and bake for an additional 10 minutes, or until the chicken is cooked through (internal temperature reaches 165°F/74°C) and the cheese is melted and bubbly.
Remove from oven and let it rest for 5 minutes before serving.