Prepare the grill for direct cooking over medium heat (350° to 450°F).
In a food processor or blender whirl the dressing ingredients until smooth and emulsified. Season with ½ teaspoon salt and ¼ teaspoon pepper.
Lightly brush the chicken breasts on both sides with oil and season evenly with salt and pepper. Brush the cooking grates clean. Cook the chicken over direct medium heat, with the lid closed as much as possible, until the meat is firm to the touch and opaque all the way to the center, 8 to 12 minutes, turning once or twice. Remove from the grill and let rest for 3 to 5 minutes. Cut the chicken into bite-sized pieces.
Fill a small saucepan with water, bring to a boil, and add the sugar snap peas. As soon as the water returns to a boil the peas should be crisp-tender. Drain immediately and run under cold water to stop the cooking process. In a large bowl combine all of the salad ingredients with the chicken pieces. Toss with enough dressing to coat the ingredients lightly. Divide the salad among four plates. Serve right away with the remaining dressing on the side.
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