Preheat oven to 425º. Season chicken breasts with garlic, paprika, salt and pepper on all sides. Using a sharp knife, slice through the side of chicken breasts leaving 1-2 inches on each side to create a pocket.
Place broccoli florets in a microwave-safe bowl with 2 tablespoons of water. Cover with a paper towel and microwave for 1 minute. Drain water and add broccoli to a medium mixing bowl with cream cheese, cheddar cheese, and dried thyme. Mix until combined.
Divide mixture between chicken breasts using a spoon and secure with 1-2 toothpicks. Heat olive oil in a large cast-iron pan over medium heat for 2 minutes. Add 2 chicken breasts to pan and sear for 3 minutes on each side. Set aside on a plate. Repeat with the remaining 2 chicken breasts. Arrange all chicken breasts in pan. Cover with aluminum foil and transfer to the oven. Bake 20 minutes. Carefully remove aluminum foil and bake 5-10 minutes longer or until chicken is cooked through. Remove toothpicks before serving.
While chicken bakes, rinse the rice in a mesh strainer to remove some of the starch. Heat butter over medium heat in a medium sauté pan. Add bell pepper, carrots, peas, onions, and rice. Stir in chicken broth and bring to a boil. Stir and cover. Reduce heat to medium-low and simmer rice for 15-20 minutes, until rice is tender and the broth is absorbed. Gently fluff rice with a fork. Season with salt, pepper, and parsley. Serve chicken hot alongside rice.