Start boiling a small pot of water. While you wait for it to heat up, trim the ends of 8 green beans and cut 1 carrot into batons that match the size of the green beans and the width of the beef. Boil for 3 minutes and then transfer to a bowl of cold water to prevent them overcooking.
Stretch a piece of thinly sliced beef out on a flat surface and sprinkle the top with flour. Place 2 green beans and 2 carrot battons stacked into a 2x2 pattern on one side of the beef.
Tightly roll the beef to encase the vegetables. Repeat with the rest of the beef slices and vegetables, placing each finished roll in a container with the seam facing down to prevent it from unrolling.
Start heating a pan on medium. Sprinkle the beef rolls with a thin even coat of flour.
Mix 2 tbsp Japanese soy sauce (koikuchi shoyu), 2 tbsp sake, 2 tbsp mirin and 1 tsp sugar in a small bowl and set by the stove ready for later.
Once the pan is hot, pour a drizzle of oil and add the beef rolls. Fry until browned on all sides.
Pour the sauce into the pan and continue to cook until thickened, turning the beef rolls occasionally to ensure they're evenly covered and don't burn. Once coated, remove the pan from the heat.