Combine all marinade ingredients in a bowl or plastic container. Trim the scallions into six inch lengths. Wash the scallions well and while still wet, wrap in a clean paper towel. Place damp towel containing scallions in the microwave for 30 seconds. Remove from microwave, open paper towel and spread the scallions out to cool. They should turn bright green and soften enough to be pliable. If you don’t have a microwave you could blanch them in boiling water for 45 seconds and then plunge into ice water and set aside.
Lay out your thinly sliced steaks (you could also use cube steaks or flank steak pounded thinly), overlapping if necessary and trim to about 6″ by 6″ squares. Lay about 5 scallions in the center of each and roll them up tightly. You can use twine or extra scallions to tie them so they don’t unravel.
Place the rolls in your marinade along with the meat scraps which can be sauteed and used in a stir fry or wrapped in lettuce leaves. Marinate for an hour or more.
Remove rolls (and meat scraps which can be cooked separately) from the marinade and pat dry.
Heat oil in a medium saute pan until hot and then place rolls into the pan. Cook for about 4 – 6 minutes until medium rare inside (or longer if you prefer), turning frequently so all sides are nicely browned. Remove from pan and let rest for a couple of minutes before slicing.
Meanwhile, strain the remaining marinade to get out the garlic and ginger pieces and add the marinade to the pan. Cook for a couple of minutes or until reduced to a syrupy consistency (don’t overdo it or it will harden when it cools!).
Cut the rolls on a diagonal and garnish with the sauce.