In a blender, combine the basil, chillies, garlic, brown sugar, fish sauce, pepper and lime juice. Blend until well incorporated.
Season the steak with salt and pepper and barbecue to medium rare, about 4 minutes a side. You can also do this under the grill. Remove to a plate, cover with foil and let rest 10 minutes. Thinly slice the steak across the grain.
In a mixing bowl, combine the steak, any steak juices from the plate and all the remaining ingredients except the lettuce. Toss with the dressing.
To serve, arrange the lettuce leaves on individual plates and mound the beef salad on top.