Remove chicken tenderloin- The chicken tenderloin is the small piece of meat attached to the underside of each breast half.
Also remove any white or yellow fat.
Put the flour in a shallow dish. Dry chicken with paper towels and season with salt and pepper. Dredge 1 chicken breast at a time in the flour, shaking off the excess.
Heat 2 tablespoons of the oil in a 12-inch skillet over medium-high heat until just smoking. Add the chicken and cook until well browned on one side, 6 to 8 minutes.
Flip the chicken over, reduce the heat to medium-low, and continue to cook until the thickest part of the breast registers 160 to 165 degrees on an instant-read thermometer, 6 to 8 minutes longer. Transfer the chicken to a serving platter, tent loosely with foil, and let rest while making the sauce.
Add 2 more teaspoons of the oil (if needed) and the garlic to the skillet and cook over medium-high heat until fragrant, about 30 seconds. Stir in the tomatoes and olives, scraping up any browned bits. (If necessary, add enough of the water to help loosen the browned bits from the bottom of the pan.) Cook until the tomatoes are just softened, about 2 minutes. Stir in any accumulated chicken juices and season with salt and pepper to taste.
Off the heat, stir in the remaining 2 tablespoons olive oil and pour the tomatoes over the chicken. Sprinkle with the feta and mint. Serve.