Combine first 5 ingredients in a large bowl, stirring well with a whisk. Add honeydew and next 4 ingredients (through green onions); toss well to coat.
Place 2 cups melon mixture on each of 4 plates; top each serving with 1/2 cup chicken. Sprinkle 1 tablespoon cilantro over each serving; top each with 1 1/2 teaspoons walnuts. Serve immediately.