Sprinkle a pinch of the salt & pepper to both sides of the chicken. In a large sauté pan, heat the oil and butter over a medium-high heat. Add the chicken pieces to the pan and sauté just until browned on both sides, about 5 minutes per side. If all the chicken does not fit in the pan, sauté it in batches. Transfer the chicken to a plate and set aside.
Add the chopped red bell pepper, onion, and carrot to the same pan and sauté over medium heat until the veggies are tender. You may need to add a drizzle more of olive oil, if the pan is dry. Add the garlic and sauté until fragrant, about 1 minute.
Add the wine and simmer until reduced by half, about 2-3 minutes. Add the whole San Marzano tomatoes with their juice, tomato paste, broth, capers and oregano, remaining salt & pepper and combine.
Return the chicken pieces to the pan and turn them to coat in the sauce. Bring the sauce to a simmer. Continue simmering over medium-low heat until the chicken is just cooked through, about 20-30 minutes. Check internal temperature with an instant read thermometer for doneness. It should read 170° F in the thickest part of the chicken.
Transfer the chicken to a platter. Boil the sauce until it thickens slightly, about 3-5 minutes. Spoon the sauce over the chicken, then sprinkle with the fresh parsley & olives and serve.