1. Cook carrots and peas in boiling water 2 1/2 minutes or until crisp-tender. Drain and rinse with cold water; drain. Combine carrot mixture, lettuce, watercress, radishes, and parsley in a large bowl; set aside.
2. Place chicken between 2 sheets of plastic wrap; pound to 1/4-inch thickness using a meat mallet or small heavy skillet. Sprinkle chicken with 1/4 teaspoon salt and 1/8 teaspoon pepper. Heat 2 teaspoons oil in a large skillet, swirling to coat. Add chicken to pan; cook 4 minutes on each side or until done. Let stand 5 minutes; cut across the grain into slices.
3. Combine remaining 2 tablespoons oil, remaining 1/4 teaspoon salt, remaining 1/8 teaspoon pepper, vinegar, and remaining ingredients in a small bowl, stirring with a whisk. Drizzle dressing over lettuce mixture; toss gently. Arrange about 2 cups lettuce mixture on each of 4 plates; divide chicken evenly among servings.