In a small bowl whisk the dressing ingredients, except the oil. Slowly whisk in the oil to make a smooth dressing.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Place the chicken breasts in a large bowl and add ¼ cup of the dressing; turn to coat. Allow the chicken to marinate at room temperature for 15 to 30 minutes before grilling. Spoon 2 tablespoons of the dressing over the onion slices and gently turn to coat evenly.
Brush the cooking grates clean. Grill the chicken, smooth (skin) side down first, 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. Grill the onion slices with the chicken over direct medium heat until charred in spots and tender, about 8 minutes, turning once. Remove the chicken and the onions from the grill and let the chicken rest for 3 to 5 minutes.
Combine the remaining ingredients in a large serving bowl. Cut the chicken into thin strips and roughly chop the onions. Add the chicken and onions to the bowl. Add enough of the remaining dressing to coat the ingredients lightly (you may not need all of it). Mix well. Serve at room temperature.