Preheat the oven to 450 degrees F. Heat 1 teaspoon olive oil in a small saucepan over medium heat. Add the shallot and cook, stirring occasionally, until softened, about 3 minutes. Stir in the rice until well coated. Add the tomato sauce and 3/4 cup water. Bring to a boil, then reduce the heat to a simmer. Cover and cook until the liquid is absorbed and the rice is tender, about 18 minutes. Remove from the heat and set aside, covered.
Meanwhile, mix the broccoli, cheese spread, lemon zest, 1 teaspoon olive oil and a pinch each of salt and pepper in a small bowl; set aside.
Line a rimmed baking sheet with foil. Cut three deep slits across the top of each chicken breast at an angle (do not cut through). Rub the chicken with the remaining 2 teaspoons olive oil and season with salt, pepper and the paprika. Stuff the broccoli mixture into the slits and transfer to the prepared baking sheet. Roast until the chicken is just cooked through, 12 to 15 minutes.
Fluff the rice with a fork. Stir in the parsley and season with salt and pepper. Serve with the chicken.