Set up your grill for direct high heat. Brush chicken breasts with 1 tablespoon oil.
Place chicken on grill, 4 inches from heat source; cook until golden brown on that side, 4–6 minutes.
Turn chicken; continue grilling until done, 4–6 minutes more. Remove from heat.
Using a fork and a chef ’s knife, cut chicken into bite-size pieces while still hot; then toss with bell-pepper slices, pasilla-pepper slices, and onions (vegetables will soften a bit). Season with salt and pepper to taste. Chill in refrigerator 15 minutes.
Meanwhile, in a small bowl, whisk together garlic, red-pepper flakes, red-wine vinegar, balsamic vinegar, and remaining 5 tablespoons olive oil. Season with salt and pepper to taste.
Add to chicken/pepper mixture and toss well. Chill in refrigerator at least 30 minutes, up to one day.
To serve, toss chicken/pepper mixture with tomatoes. Garnish with olives and basil leaves.