In a glass or ceramic baking dish, whisk 2 tbsp. EVOO, 2 tbsp. soy sauce, the garlic and half the ginger. Add the steak; turn to coat. Let marinate 15 minutes, turning halfway through.
In a large bowl, whisk together the rice vinegar, 2 tbsp. EVOO and the remaining soy sauce and ginger. Toss with the sliced cabbage, carrots and basil; season with salt and pepper.
In a large cast-iron or other heavy skillet, heat the remaining 1 tbsp. EVOO over medium. Remove the steak from the marinade and cook to desired doneness, about 4 minutes per side for medium-rare. Transfer the steak to a cutting board, tent loosely with foil, and let rest 5 minutes.
Transfer the salad to a platter. Thinly slice the steak against the grain. Fan out the sliced steak on top of the salad.