Remove back bone and all other bones from each chicken except wing bones; flatten or flatten boneless chicken breast.
Generously rub each with spice rub; reserve.
Season 1 chicken with salt; grill, turning occasionally, until skin is crisp and flesh is just firm to the touch, about 5 minutes meanwhile, cut 1 avocado into wedges; sauté in a little oil.
To assemble the dish: pour ¼ cup heated tomatillo sauce in the center of a large dinner plate; layer with ⅓ cup heated grilled vegetables, sautéed avocado, and ¼ cup heated cumin-scented black beans.