1. Rinse chicken and pat dry. Sprinkle breast halves lightly all over with salt and pepper. Heat 1 tablespoon olive oil in a 10- to 12-inch nonstick frying pan over medium heat. Add chicken and cook, turning once, until browned on both sides but still slightly pink in the center of the thickest part (cut to test), about 8 minutes total. Transfer chicken to a plate.
2. Pour remaining 1/2 tablespoon olive oil into pan. Add shallots and garlic and stir often until fragrant but not browned, 1 to 2 minutes. Add peppers and stir frequently until slightly soft, 1 to 2 minutes. Add vinegar, olives, capers, and the chicken. Cover, lower heat to medium-low, and cook until peppers are soft and chicken is no longer pink in the center of the thickest part (cut to test), 5 to 8 minutes longer. Add salt and pepper to taste. Transfer each breast half to a plate and spoon peppers and juices evenly over warm chicken.