Trim the chicken of all fat and, if pieces are thicker than 3/8 of an inch, pound them between sheets of plastic or waxed paper until they are more uniformly even. Combine half of the lime juice with the ground cumin and salt and rub this over the chicken. Preheat the broiler or grill. Start to grill the chicken and cook for about 6 minutes a side or until just cooked through. Combine the avocados with the mangoes, cucumbers and remaining lime juice. Season with salt and pepper. When the chicken is cooked through, remove the pieces to a cutting board to cool slightly and, in a blender or food processor, puree the vegetable oil, cilantro, scallion with white wine vinegar or Verges and season well with salt and pepper. To serve, cut the chicken, across the grain, on a diagonal, into thin slices. Spoon about 1 1/2 cups of mango and avocado mix on each dinner plate. Center the chicken slices over the mango salad and spoon the cilantro vinaigrette over the chicken. Serve at room temperature.