Mix the soy sauce, brown sugar, sesame oil, and garlic in a bowl. Add the shaved beef, toss well, and let marinate for 15 minutes.
Melt the butter in a skillet over medium heat and toast the hoagie rolls face down until golden. Remove and set aside.
Turn heat to high, add the sliced onions, and cook for 2 minutes. Add the marinated beef in an even layer and sear undisturbed for 2 minutes.
Toss the drained cabbage into the skillet with the beef and cook until the meat is browned completely.
Divide the meat into two portions in the pan. Top each with two slices of cheese, cover the pan with a lid, and turn off the heat for 1 minute.
Scoop the melted cheesy meat portions directly onto the toasted hoagie rolls and serve immediately.
