Prepare grill to medium-high heat.
Place chicken between 2 sheets of heavy-duty plastic wrap; pound each piece to 1/2-inch thickness using a meat mallet or small heavy skillet. Sprinkle chicken with parsley, tarragon, 1/4 teaspoon salt, and 1/4 teaspoon pepper.
Place chicken on grill rack coated with cooking spray; grill 4 minutes on each side or until done. Transfer to a plate; cool to room temperature.
Combine oil, vinegar, remaining 1/4 teaspoon salt, and dash of pepper in a small bowl, stirring with a whisk.
Combine arugula, greens, apricots, and onion in a large bowl. Pour vinaigrette over arugula mixture; toss well to coat. Place about 2 cups arugula mixture on each of 4 plates. Cut chicken breast halves crosswise into thin slices; top each serving evenly with chicken. Serve immediately.