In a 9-by-13-inch glass baking dish, combine soy sauce, fish sauce, and sugar, Add steak, turning to coat, and marinate at room temperature 15 minutes (or in refrigerator, overnight).
Heat a large cast-iron skillet over high. Remove steak from dish and reserve marinade. Add steak to skillet, reduce heat to medium-high, and cook, until dark brown, 3 to 5 minutes per side for medium-rare, flipping once. Add reserved marinade to skillet and cook 15 seconds. Turn steak to coat in syrupy marinade. Transfer steak to a cutting board and let rest 5 minutes. Slice thinly against the grain.
In a large bowl, toss together lettuce, cucumber, tomatoes, carrots, mint, and basil. Add dressing and steak and toss to combine. Serve immediately.