Split each breast piece in half parallel to the cutting board, making thin cutlets. Cover with plastic and pound cutlets to an even 1/4" thickness.
Whisk together lemon juice, oil, salt and pepper. Pour half over the chicken; set the other half aside. (You can proceed immediately, or let the chicken marinate in the refrigerator up to one hour for more flavor.).
Heat a grill or grill pan to high heat. Brush with more oil, then place chicken on grill/grill pan. Sprinkle exposed side evenly and lightly with cumin. Cook 1-2 minutes on each side, until browned and opaque.
Trim any tough stems off the watercress. Trim any tails off the tip of the radishes, and trim off the greens, leaving about 1/4". Slice radishes from pole to pole, as thin as possible. Toss radishes with watercress, avocado and reserved dressing.
Serve chicken with salad on the side.