For the chicken: In a shallow pan large enough to hold chicken in a single layer, whisk together oil, vinegar, and mustard; add chicken, turning to coat. Cover and refrigerate for at least 2 hours.
Heat the grill to high; lightly oil the grates. Lift chicken from marinade and discard any remaining marinade. Grill until fully cooked and opaque throughout, 6 to 8 minutes per side.
For the dressing: In a blender combine mango, cilantro, mint, curry powder, and vinegar. Puree until smooth. With the motor running, add oil; season with salt and pepper.
To serve: Slice chicken crosswise. Arrange on a platter or four serving plates, along with avocado, watercress, and onion; serve the dressing on the side.