Preheat the oven: Set to 375°F (190°C). Lightly grease a 9x13-inch baking dish with olive oil or nonstick spray.
Season the chicken: Pat the chicken dry. Sprinkle both sides with 1/2 teaspoon salt, 1/4 teaspoon pepper, and 1/2 teaspoon paprika.
Set aside.
Build the base: In the baking dish, whisk together the chicken broth, milk, and cream of chicken soup until smooth. Stir in the rice, onion, garlic, thyme, remaining salt and pepper, and remaining paprika. Fold in peas and carrots.
Nestle the chicken: Place the seasoned chicken pieces on top of the rice mixture, gently pressing them in so they’re partly submerged but still visible.
Add butter: Drizzle the melted butter over the chicken and rice for richness and color.
Cover and bake: Cover the dish tightly with foil.
Bake for 35 minutes.
Uncover and finish: Remove the foil, sprinkle cheese on top if using, and bake another 15–20 minutes, until the rice is tender and the chicken reaches an internal temperature of 165°F (74°C).
Rest and garnish: Let the casserole rest for 10 minutes. This helps the rice finish steaming. Sprinkle with fresh parsley and adjust seasoning to taste.