Place steak on large plate and freeze until firm, about 30 minutes.
Combine ½ cup soy sauce, ¼ cup honey, ¼ cup mirin, and ¼ cup sake in a small saucepan over medium. Bring to a simmer, stirring to combine, and cook until slightly syrupy and reduced by half (about ⅔ cup), 8–10 minutes.
Off heat, stir in 1 tablespoon sesame seeds and ¾ grated ginger, if using. Divide sauce evenly between 2 bowls and let cool; set one bowl aside for serving.
Transfer steak to a cutting board and, starting at narrow tapered end, slice steak ⅜-inch thick on bias against grain until width of steak is 7-inches (depending on size of steak, you will need to remove 2–3 slices until steak measures 7-inches across).
Cut steak in half lengthwise then continue to cut each half (on a bias and against the grain) into ⅜-inch thick slices. You should have at least 24 slices. Pound each slice to 3/16-inch thickness using a flat meat tenderizer or rolling pin.
Arrange 3 slices of meat on cutting board with short side of slices facing you, overlapping slices by ¼-inch and alternating tapered ends to form a rough rectangle 4–6-inches wide and at least 4-inches long.
Place 4 scallion halves along edge of rectangle parallel to you and perpendicular across steak slices, with white tips slightly hanging over edges of steak on either side. Starting from bottom edge (nearest you), roll steak over scallions (away from you), into a tight roll. Insert 3 toothpicks, equally spaced amongst each slice of meat, into meat ends and through center of roll.
Transfer roll to platter and repeat with remaining steak and scallions. (Assembled rolls can be refrigerated for up to 24 hours.)