1.Combine panko and milk in a large mixing bowl then set aside. Heat 1 teaspoon of oil in a nonstick pan over medium heat, then add onion. Lightly season with a dash of salt and sauté for 2-3 minutes or until the edges are golden brown. Set aside to cool completely.
2.In the bowl with panko add beef, sautéed onions, garlic, egg, 2 tablespoons ketchup, Worcestershire sauce, 1/2 teaspoon salt and pepper. Mix with your hands until evenly combined.
3.Scoop a quarter portion of the meat and toss back and forth between your hands before forming into a tight patty and place on a plate. Repeat three more times.
4.Heat 2 tablespoons of oil in the same pan over medium heat. Place the patties on the pan, indent the center, and cook for about 2-3 minutes on each side then transfer to a plate.
5.In the same pan melt butter over medium heat. Add flour and stir to combine. Once it turns light brown, add mirin, sugar, soy sauce, 2 tablespoons ketchup, and water. Cook for about 2 minutes until the sauce is thickened.
6.Pour the sauce over the patties and serve immediately with a side of mashed potatoes and vegetables.