In a Dutch oven or heavy-bottomed pot, heat the olive oil over medium heat and sauté the onions and peppers for 10 to 15 minutes or until translucent. Add the garlic, chili powder, cumin, red pepper flakes, cayenne and salt. Cook for 1-2 minutes, until fragrant. Add the canned tomatoes and chicken stock to the pot. Bring to a boil, then reduce the heat and simmer, uncovered, for 30 minutes, stirring occasionally.

Preheat the oven to 350 degrees F. Rub the chicken breasts with olive oil and place them on a baking sheet. Sprinkle generously with salt and pepper. Roast the chicken for 35 to 40 minutes, until a meat thermometer registers 165 degrees F. Let cool slightly. Separate or shred the meat from the bones and skin, or cut into chunks. Add the chicken to the chili and simmer, uncovered, for another 20 minutes. To finish the soup add a splash of sherry vinegar and/or lime juice and sprinkle on the fresh chopped basil. Serve the chili with the toppings of your choice.