Heat 2 tablespoons of the oil in a large Dutch oven over high heat. Sprinkle the chicken pieces on both sides with salt and pepper. Working in batches, place the chicken in the pan, skin-side down, and cook until golden brown, 3 to 4 minutes. Turn the chicken over and cook until golden brown, 2 to 3 minutes. Remove the chicken to a large plate.
Add another tablespoon of the oil to the pan and heat until almost smoking. Add the mushrooms, season with salt and pepper and cook, stirring occasionally, until they are golden brown and their liquid has evaporated, about 10 minutes. Remove to a plate.
Add the remaining 1 tablespoon oil, the onions and the bell peppers to the pan and cook until soft, about 4 minutes. Add the garlic, serrano chiles and red pepper flakes, and cook for 1 minute. Stir in the wine and cook until almost completely reduced. Add the chicken broth, tomatoes, rosemary and thyme, and bring to a simmer. Return the dark meat and mushrooms to the pan, reduce the heat to medium-low, cover the pan and cook until the chicken is tender, about 30 minutes.
Add the white meat to the pan, cover and continue cooking for 10 minutes. Remove the chicken with a slotted spoon to a large shallow bowl and tent it loosely with foil.
Raise the heat under the pan to high and cook the liquid, stirring occasionally, until thickened to a sauce consistency, about 15 minutes.
Season with salt, pepper and honey. Stir in the basil and capers, and pour the sauce over the chicken. Garnish with fresh basil sprigs. Serve over spaghetti and sprinkle with shaved Parmesan.
From Food Network Kitchens; after further testing and to ensure the best results this recipe has been altered from what was in the actual episode.