Heat 2 tablespoons of the oil in a large Dutch oven over high heat. Season the chicken pieces on both sides with salt and pepper and place in the pan, skin side down and cook until golden brown, 3 to 4 minutes. Turn the breasts over and cook until golden brown, 2 to 3 minutes. Remove the chicken to a large plate.
Add the remaining oil to the pan and heat until almost smoking. Add the mushrooms and cook until golden brown, season with salt and pepper and remove to a plate.
Add the onion and bell pepper to the pan and cook until soft, 3 to 4 minutes. Add the serrano, garlic and chili flakes to the pan and cook for 1 minute. Stir in the wine and cook until almost completely reduced. Add the chicken stock, tomatoes, and rosemary and bring to a simmer. Return the chicken and mushrooms to the pan, reduce the heat to medium-low, cover the pan and cook until the chicken is tender, about 30 minutes. Remove the chicken with a slotted spoon to a large shallow bowl and tent loosely with foil.
Increase the heat to high and cook the sauce, stirring occasionally, until slightly thickened, 8 to 10 minutes. Season with salt and pepper, stir in the vinegar and basil and pour the sauce over the chicken. Garnish with fresh basil sprigs. Serve over spaghetti and sprinkle with grated cheese.