Whisk together vinegar, garlic, mustard, mint, 1/4 teaspoon of the salt and 1/8 teaspoon of the black pepper. Gradually drizzle in olive oil, whisking constantly, so mixture emulsifies.
In a medium bowl, combine tomatoes, basil and 3 tablespoons of the dressing. Cover and set aside.
Heat a gas grill to medium-high or the coals in a charcoal grill to medium-hot. Lightly coat grill rack with oil or nonstick cooking spray.
Brush chicken with remaining dressing and season with remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Grill 5 to 6 minutes per side or until internal temperature reaches 160 degrees. Scatter cheese over top during last minute of cooking.
Toss cooked pasta with half of the tomato salad. Serve with chicken and remaining tomato salad.
