Japanese-Style Omelette Burger

I've read about but not made nor eaten an American hamburger omelette. However, when the dish appears in a currently airing Japanese drama (dorama), A Girl and Three Sweethearts, I noticed that it is not served with bread but in a shallow pool of demi-glace. That's interesting.

Japanese-style hamburgers [and other Asian burgers] are a bit different because the meat is seasoned with a little onion and sake and softer like meatloaf with the addition of bread crumbs, milk, and egg. The Japanese burger steaks are usually served with a sauce made with a combination of ketchup and wooster sauce.

I made a few [Filipino- style] burgers and wrapped one in a 3-egg omelette but cheated on the demi-glace sauce. I used a little of my homemade beef broth, some ready made beef powder, red wine, and homemade ketchup for the demi-glace. Oishii!

Japanese-Style Omelette Burger

burgers (my version)

¼ cup panko bread crumbs

2 tablespoons milk

1 pound 80/20 ground beef

1 teaspoon onion powder

½ teaspoon garlic powder, optional

½ teaspoon fine sea salt

a dash of ground black pepper

1 teaspoon sake

¼ teaspoon Worcestershire sauce

1 teaspoon liquid smoke, optional 

1 egg

omelette for 1 serving

3 eggs, well beaten

dash of salt

½ teaspoon mirin, optional

1 teaspoon butter

sauce

1 cup ketchup or tonkatsu sauce

2 tablespoons water

¼ teaspoon Worcestershire sauce

• Burgers: Moisten bread crumbs with milk. Mix in all the ingredients. Form into 4 patties and cook well done in a dry cast iron skillet. Keep warm while making the omelette. 
• Omelette: Melt butter over low-medium heat in a cast iron skillet. Stir salt and mirin into the eggs and mix well. Cook omelette until bottom is dry but top is still moist. Place a cooked hamburger in the middle of the omelette and fold on all sides towards the center. Flip omelette burger onto a plate or a small cast iron skillet. 
• Sauce: Heat ketchup, water, and Worcestershire sauce; adjust thickness with more water or ketchup. (If you have the time, make a proper demi-glace.)

• Omelette Burger:  Pour ¼ cup to ½ cup of sauce around the omelette. Serve while warm.