Put each breast half between 2 sheets of plastic wrap. Flatten to a uniform thickness of 1/4 inch; set aside. Mix the vinegar, mustard, and seasonings in a small dish. Combine the tomatoes, green onions, mint, cheese, 1-1/2 teaspoons of the oil, and 1-1/2 teaspoons of the vinegar mixture in a small bowl. Set aside.
Place a large skillet over high heat. When hot, add the remaining 1-1/2 teaspoons oil and heat about 30 seconds. Add the chicken breasts. Cook, turning once, until they are cooked through, about 3 minutes total. Transfer the chicken to two dinner plates.
Add the tomato mixture to the skillet, cook, and stir about 20 seconds and divide between the chicken breasts. Serve at once.