Preheat the oven to 375°F (190°C).
In a medium saucepan, bring the chicken broth to a boil. Add the rice, reduce heat to low, cover, and simmer for 15 minutes or until the rice is tender and liquid is absorbed. Remove from heat and let cool slightly.
While the rice cooks, heat olive oil in a skillet over medium heat. Add the chopped onion, carrot, and celery. Sauté for 5-7 minutes until vegetables are soft.
Add minced garlic to the skillet and cook for 1 minute more until fragrant. Remove from heat.
In a large bowl, combine the cooked rice, sautéed vegetables, mozzarella cheese, Parmesan cheese, chopped parsley, dried thyme, salt, and black pepper. Mix well.
Using a sharp knife, cut a horizontal pocket into each chicken breast, being careful not to cut all the way through.
Stuff each chicken breast pocket evenly with the rice mixture. Secure the opening with toothpicks if needed.
Place the stuffed chicken breasts in a baking dish. Sprinkle paprika over the tops of the chicken breasts.
Bake in the preheated oven for 30-35 minutes, or until the chicken is cooked through and the internal temperature reaches 165°F (74°C).
Remove toothpicks before serving. Let the chicken rest for 5 minutes before slicing and serving.