Heat a medium size saucepot over medium heat with 1 tablespoon of extra-virgin olive oil, about 1 tablespoon, and the butter. Add 1/4 of the chopped onion and season with salt and pepper, cook for about 1 minute, add the rice and stir to coat in the oil. Add 2 1/2 cups of chicken stock to the rice and bring up to a boil. Cover the pot, reduce heat to simmer and cook rice for 15 to 18 minutes or until tender and cooked through.
While the rice is cooking, start the chicken. Preheat a large skillet over medium-high heat with about 2 tablespoons of extra-virgin olive oil. Season the chicken breasts with smoked paprika, salt, pepper and the thyme; add the seasoned chicken to the skillet. Cook breasts 5 to 6 minutes on each side or, until cooked through.
Remove the chicken from the skillet to a plate and loosely cover with aluminum foil. Return the skillet to the stove top and over medium-high heat add chorizo, cook for 2 minutes stirring frequently. Add the remaining onion, garlic, grated carrot, and roasted red pepper to the chorizo and season with a little salt and pepper. Cook until the onions are slightly tender, about 2 to 3 minutes. Add 1/2 cup of chicken stock, scrape up any bits that have stuck to the bottom of the pan. Add the frozen peas and parsley; continue to cook for 1 to 2 minutes to heat the peas through. Taste and adjust seasonings with a little salt and pepper.
To serve, arrange the chicken breasts on dinner plates and top with lots of sauce. Fluff the rice with a fork and serve alongside.