When cutting the chuck steak, you want the meat to be rather thin so that it's easy to eat. Butterfly the chuck steak and tenderize by hitting the meat with the blunt edge of a heavy knife. You can also slice the meat thinly, but hitting it flat will make the meat easier to chew.
Marinate the meat with garlic, cooking wine, brown sugar, and shoyu for at least 30 minutes. Leave this in the fridge.
Ready a large pan on high heat with ¼ cup of cooking oil. Add the flour into a bowl and dip the meat into the bowl one by one. Make sure the flour covers the meat all over so that the egg will stick to the meat.
Add the eggs into a separate bowl and scramble them. Dip the floured meat completely into the eggs so that they stick onto the flour. Transfer to the pan.
It will only take a few minutes to cook the meat jun because the meat is so thin and eggs cook fast. Adjust the heat as needed and add oil when necessary while cooking.
Flip once and cook evenly until both sides are brown. The outer egg layer will not be a bright yellow because of the beef and marinade. Meat jun can be served as is, but in Hawaii we typically see this sliced to bite-sized pieces so that they're easy to eat and dip into the sauce.