1Place chicken in shallow glass or plastic dish. Brush 2 tablespoons of the pesto over tops of chicken; turn chicken. Brush with 2 tablespoons of the pesto; sprinkle with salt. Cover and refrigerate at least 1 hour but no longer than 24 hours.
2Heat coals or gas grill for direct heat. Grill chicken uncovered 4 to 6 inches from medium heat 15 to 20 minutes, turning once, until juice is no longer pink when centers of thickest pieces are cut. Add bell pepper for last 10 minutes of grilling until bell pepper is crisp-tender. Add bread for last 3 minutes of grilling, turning once until toasted.
3Cut bell pepper into strips. Spread remaining 1/2 cup pesto on one side of bread slices. Place chicken on 4 bread slices; top with bell pepper, basil and remaining bread.