Place the steak in a large food storage bag. Combine the soy sauce, Mirin, brown sugar, sesame oil, garlic, green onions, and ginger in a mixing bowl. Whisk to combine. Pour the marinade over the meat and let sit at room temperature for 30 minutes. (If making ahead, you can marinate it in the refrigerator for up to 24 hours.)
Heat a grill or grill pan to medium-high heat.
Pat the steak dry with paper towels and grill 2 minutes per side. Let rest a few minutes then slice thinly.
Grill the tortillas 30-60 seconds per side. Keep warm.
To serve, place a lettuce leaf on top of a tortilla, top with some sliced meat and some quick pickled cabbage slaw.