In a 4- or 5-quart pot, heat the olive oil over medium heat until hot and rippling. Add the onions and reduce the heat to medium-low, cooking for 10-12 minutes until the onions are very soft. Add the garlic and cook for one minute, stirring constantly. Add the bell peppers, chili powder, cumin, red pepper flakes, cayenne, salt, and dried basil (if using fresh, you'll add it later). Cook for 1-2 minutes.
Crush the tomatoes by hand or in a blender or food processor. They don't need to be creamed to a pulp - just six or so pulses to get them blended a bit. Add to the pot with the fresh basil, if using.
Bring to a boil, then reduce the heat and simmer, uncovered, for 30 minutes, stirring occasionally. Add the cooked chicken and stir. Simmer, uncovered, for another 20 minutes. The chili will thicken and the liquid will reduce slightly. Serve with any additional toppings (chopped onions, corn chips, grated cheddar, sour cream, avocado, etc.).
Notes
Chicken: I use this method for cooking the chicken. Spiciness: adjust the dried red pepper flakes and the cayenne pepper according to your desire for spiciness. The amount in the recipe is perfect for us, enough heat to make it interesting but not so much that my kids are dying. Tomatoes: I've made this recipe with diced tomatoes and crushed tomatoes and let me tell you, blending up the whole plum tomatoes is the way to go. More tomato flavor, baby.