Place each chicken breast half between two pieces of plastic wrap. Using the flat side of a meat mallet, pound chicken lightly to about 1/2-inch thick. Discard plastic wrap. Brush 1 tablespoon oil over both sides of chicken; sprinkle chicken lightly with salt and pepper. Heat a grill or grill pan coated with nonstick cooking spray over medium heat. Grill 4 minutes on each side or until done (165 degrees F).
Meanwhile, in a small bowl combine the remaining oil, basil, onion, garlic, vinegar, and lemon juice. Season to taste with salt and pepper.
Transfer chicken to serving plates. Top with basil mixture. Let stand 5 minutes.