Toss together the chicken, cubed melon, cucumber, zucchini, and green onions in a large mixing bowl.
For dressing, combine lime juice, salad oil, water, cilantro or mint, sugar, and white pepper in a screw-top jar. Cover and shake well. Drizzle 1/2 cup of the dressing over the chicken and melon mixture. Toss lightly to coat. Chill remaining dressing for another use.
Divide melon mixture among melon halves. If desired, garnish with fresh cilantro and frisee. Serve immediately. Makes 4 servings.