1Heat gas or charcoal grill. In small bowl, mix 2 teaspoons vegetable oil and the soy sauce; brush on both sides of chicken. Brush grill rack with vegetable oil. Place chicken on grill. Cover grill; cook over medium heat 15 to 20 minutes, turning once, until juice of chicken is clear when center of thickest part is cut (170°F).
2Meanwhile, in small bowl, beat all dressing ingredients with wire whisk.
3In large bowl, toss all salad ingredients with dressing; divide among 6 serving plates. Cut chicken diagonally into 1/2-inch slices. Top salads with chicken.