In a large saucepan, melt 3 tablespoons butter over medium heat. Add the pasta and toast for 3 minutes. Stir in the rice to coat. Add 2 cups chicken stock and bring to a boil, then cover the pot, lower the heat and simmer for 15 minutes; fluff with a fork. (If the rice sticks near the end of the cooking time, add a splash more stock.) Season with salt and pepper.
In a large, deep skillet with a tight-fitting lid, heat 1 tablespoon EVOO, 1 turn of the pan, over medium-high heat. Add the chorizo and cook, turning, until browned, 2 to 4 minutes. Using a slotted spoon, transfer to a paper-towel-lined plate. Season the chicken with a lot of pepper and a little salt and add to the drippings in the skillet. Cook over medium heat, turning once, until browned, about 15 minutes. Transfer to a plate
Add the remaining 1 tablespoon EVOO to the skillet, 1 turn of the pan. Add the onions, season with salt and pepper and cook, tossing frequently, for 2 minutes. Add the lemons, olives, garlic, bay leaves and paprika; cook, stirring frequently, for 2 minutes. Add the wine and cook, scraping up the browned bits from the pan. Stir in the remaining 1 cup chicken stock.
Return the chicken and chorizo to the skillet, cover the pan, lower the heat and simmer for 5 minutes. Transfer all the solid ingredients to a platter. Reduce the sauce for a minute over high heat, then stir in the remaining 1 tablespoon butter and the parsley. Pour the sauce over the chicken. Discard the bay leaves. Serve alongside or on top of the pilaf.
