Oil grill and heat barbecue to medium-high. Pour 3 tbsp. into a shallow dish. Add steaks. Turn to coat. Tear lettuce into pieces and place in a large bowl. Cut cucumber in half lengthwise, scrape off seeds with a spoon. Thinly slice. Slice mango cut into strips. Add to lettuce along with carrots, radishes, chives. In a small bowl, whisk lime juice with both oils, remaining teriyaki and fresh ginger, if using. Add salt if needed.
Barbecue steaks, uncovered, until they give a slight resistance when pressed, 5 to 6 min per side for a steak 1" thick, if you like them medium-rare. Remove to a cutting board. Cover with foil. Let stand 5 min. Thinly slice. Toss half the dressing with salad. Serve salad topped with steak slices. Drizzle with remaining dressing.