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 just cooked.
While the chicken is cooking, cook the onions in the oil over medium-low heat for 10 to 15 minutes, until translucent.
While onions are cooking, use a pairing knife to pierce and slightly break open the tomatoes to release their water and seeds. Crush the tomatoes by hand, or in batches in a food processor fitted with a steel blade (pulse 6 to 8 times), or simply chop well with a sharp knife. Set aside.
To the onions, add the garlic and cook for 1 more minute. Add the bell peppers, chili powder, cumin, red pepper flakes, cayenne, and salt. Cook for 1 minute.
Add the basil, bring to a boil, then reduce the heat and simmer, uncovered, for 30 minutes, stirring occasionally.
While chili is simmering, separate the chicken meat from the bones and skin and shred, or cut it into 3/4-inch chunks. (All a matter of preference.) Add the chicken to the chili and simmer, uncovered, for another 20 minutes. Taste for seasoning.
Serve with the toppings, or refrigerate and reheat gently before serving.