Blanch snow peas in boiling, salted water for 30 seconds, until just tender. Drain and rinse immediately in cold water. Drain again.
Combine ginger and vinegar in a small bowl. Gradually whisk in oil and season to taste.
Gently toss together melon slices, snow peas, shredded chicken and ginger dressing. Top salad greens with chicken and melon salad. Sprinkle with snipped tarragon, if desired.