Arrange all vegetables on large rimmed baking sheet. Brush vegetables lightly on both sides with 1/3 cup Mango-Sesame Dressing; sprinkle with salt and pepper.
Arrange chicken on sheet of foil. Brush both sides of chicken with 1/3 cup dressing, then sprinkle with salt and pepper.
DO AHEAD: Vegetables and chicken can be prepared 2 hours ahead. Let stand at room temperature.
Coat grill rack generously with nonstick spray and prepare barbecue (medium-high heat). Grill vegetables until just tender, turning occasionally, about 8 minutes for onion rounds and 4 minutes for mushrooms, radishes, bok choy, and pepper strips. Return all vegetables to same baking sheet.
Grill chicken until cooked through, 5 to 6 minutes per side. Transfer chicken to cutting board. Let stand 5 to 10 minutes. Cool 2 chicken breasts; wrap and chill for Asian Chicken-Noodle Salad.
Arrange remaining 4 chicken breasts and vegetables on platter. Serve with remaining dressing.
Combine all ingredients in blender.
Cover tightly; blend until dressing is smooth.
Season to taste with salt and pepper.
DO AHEAD: Can be made 1 week ahead. Cover and chill.