Whisk soy, oil, juice and ginger in bowl. Pour over tuna in baking dish. Refrigerate 15 minutes, turning once. Mix mayonnaise, ginger, soy in bowl. Refrigerate.
Prepare grill with medium-hot coals; coat rack with cooking spray. Or coat stove-top grill pan; heat over high heat until pan starts to smoke slightly, about 3 minutes.
Remove tuna from marinade; pat dry with paper toweling. Cook on grill or pan, 3 minutes a side for medium. Transfer to platter; tent with foil.
Spread bread with mayonnaise; make 4 sandwiches with lettuce, avocado and tuna.