Make the marinade. Mix together all the ingredients in a small bowl. Put the chicken breasts in a non-metallic dish and, reserving 2 tablespoons for basting, pour the rest of the marinade over the chicken, coating it thoroughly. Cover and refrigerate for 20 minutes.
Meanwhile, make the dressing. Mix together the lemon juice, soy sauce, cumin and oil in a bowl. Set aside.
Remove the chicken from the marinade, reserving the marinade, and place it on a foil-lined baking sheet. Put the baking sheet on a rack and place it about 15 cm (6 inches) below a preheated grill. Cook the chicken, turning occasionally and brushing it with the reserved marinade, for about 10 minutes or until a skewer will easily go into the chicken and the juices run clear.
Put the shredded cos and radicchio lettuces in a large bowl. Toss with 1–2 tablespoons of the dressing then arrange on a large platter. Cut the chicken breasts into 1 cm (½ inch) slices and set them on the lettuce.
Garnish with the avocado slices. Pour the remaining dressing over, sprinkle with chopped coriander and serve.