Place each chicken breast half between two pieces of plastic wrap. Using the flat side of a mallet, pound the chicken lightly to about ½-inch thick. You want all of the chicken pieces to have the same thickness for easy grilling. Brush 1 tablespoon oil over both sides of the chicken; sprinkle lightly with salt and pepper.
Heat a grill to medium. Grill 4 minutes per side or until done (165℉)
Meanwhile, for the chimichurri, in a small bowl combine the remaining oil, basil, onion, garlic, vinegar, and lemon juice. Season to taste with salt and pepper.
Serve the grilled chicken topped with the chimichurri sauce.