Pour the olive oil into a dutch oven with enamel interior, and heat over high flame. Clean the chicken and cut into pieces. Cut the peppers into pieces as well, and eliminate the seeds. Peel the shallots and cut them coarsely, then peel and crush the cloves of garlic.
Plunge a piece of bell pepper into the olive oil. When the pepper begin to sizzle, plunge half of the peppers and cover with the lid. Cook for 7-8 minutes, holding the dutch oven covered as much as possible. When the sizzle will decrease in intensity, the peppers are cooked: They should be golden brown but not burned. Drain the bell peppers from the oil and let them rest on a plate lined with paper towel. Repeat the process with the other half of the peppers.
Just after drained the bell peppers, plunge the chicken in the same oil, a few pieces at one time, with the skin side down. As soon as the skin will become crisp, turn over the chicken pieces and cook until the surface is golden brown. Drain the chicken and line over paper towels.
Save 4 tablespoons from the oil used frying. Pour 1 tablespoon of vinegar into the pot and put on low heat. When vinegar begins to simmer, gently deglaze brown food residue, then add the saved frying oil and pour the shallots and the cloves of garlic. Stir-fry the vegetables until are golden brown.
When the shallots and garlic are golden brown, add the chicken. Raise the flame to the maximum, stirring the meat. After a few minutes pour the white wine around the chicken pieces. Cook over high heat for 4-5 minutes, until the alcohol of the wine has evaporated. Now, add the crushed tomatoes, and 5 basil leaves into pieces. When the sauce begins to simmer, set the heat to minimum, cover the pot with the lid and let cook for 1 hour.
After 1 hour, our chicken stew sauce should have a brick-red color and an intense aroma. Add the bell peppers, the olives and the capers. Mix gently, then cover the dutch oven with the lid, and cook 30 minutes at minimum heat.
When the chicken stew is cooked to perfection, complete with the remaining fresh basil leaves and a generous dash of black pepper and salt to taste. All this hard work will be rewarded with a tasty and incredibly tender stew, delicious freshly cooked, superlative heated the next day!