Set the trimmed flank steak on a large plate and chill it in the freezer for about 30 minutes until firm. The meat should resist the knife slightly without becoming frozen solid.

Warm the soy sauce, honey, mirin, and sake in a small saucepan over medium heat, stirring until blended. Let the mixture simmer for 8 to 10 minutes, until reduced by half to about 2/3 cup and thick enough to coat the back of a spoon.

Take the saucepan off the heat and stir in the sesame seeds and grated ginger. Divide the glaze evenly between two bowls, then leave both bowls to cool while reserving one strictly for serving.

Move the chilled steak to a cutting board and trim slices from the narrow end until the steak measures about 7 inches across. Cut the beef on a bias against the grain into 3/8-inch slices so the strips stay broad enough to roll.

Cut the steak in half lengthwise, then continue slicing both pieces on the bias against the grain until you have at least 24 slices. Pound each slice between protective sheets until it reaches about 3/16-inch thick and bends easily without tearing.

Lay 3 beef slices on the board with their short sides facing you, overlapping each by 1/4 inch and alternating the tapered ends. The joined slices should create a rough rectangle about 4 to 6 inches wide and at least 4 inches long, with no open gaps.

Arrange 4 scallion halves along the near edge of the beef, keeping them parallel to you and perpendicular to the meat strips. Alternate thicker white sections with slender green sections so the filling stays even from end to end.

Roll the beef tightly away from you around the scallions and secure it with 3 evenly spaced toothpicks. The roll should feel compact with no air pockets or loose flap along the seam.

Repeat with the remaining beef and scallions, placing the finished rolls on a platter. Cover and refrigerate the assembled rolls for up to 24 hours if preparing them ahead.

Open the bottom grill vent fully and light a large chimney starter filled three-quarters full with charcoal briquettes. When the upper coals show a coating of ash, spread them over half of the grill and replace the grate.

Cover the grill with the lid vent fully open and heat for about 5 minutes. The grate should be very hot before the rolls go on, which helps the beef char before the glaze can scorch.

Set all gas burners to high, close the lid, and preheat the grill for about 15 minutes. Keep the burners on high so the beef gets quick, concentrated heat across each side.

Oil the hot grill grate, then place the rolls over the coals on a charcoal grill or directly over the burners on a gas grill. Set each roll with its seam facing down first so the meat edges can seal together.

Cook the first side for about 4 minutes without brushing on glaze. Wait for char marks to begin forming before turning, since early glazing can burn the honey.

Turn the rolls and brush the cooked side with glaze from the basting bowl, then cook the second side for about 4 minutes. The beef should develop a dark caramelized surface while the scallions start to steam inside.

Turn and glaze the remaining sides, cooking until all four sides are evenly charred. Check from the end of a roll with an instant-read thermometer; the scallion core should register around 150°F, with total grill time around 10 to 12 minutes.

Transfer the cooked rolls to a cutting board, tent them loosely with aluminum foil, and let them rest for 5 minutes. Discard any glaze left in the bowl used for brushing the meat.

Remove the toothpicks, cut each roll crosswise into 3 pieces, and arrange them cut side down on a clean platter. Drizzle with 2 tablespoons of the reserved glaze and serve the remaining reserved glaze alongside.