Heat about 2 teaspoons of oil in a medium non-stick skillet over medium heat. Add diced onions and sauté for 4-5 minutes until soft and slightly golden brown. Remove from heat and set aside to cool.
In a large bowl, combine ground beef, ground pork, egg, panko bread crumbs, sautéed onions, salt, and pepper. Mix thoroughly until fully combined. Knead for an additional 2-3 minutes until the meat becomes sticky and well-bound.
Divide the ground meat mixture into 6 equal portions and use your hands to shape them into patties. Press your thumb into the center of each patty to create a small indent, which helps prevent the patties from shrinking as they cook.
Heat 1 tablespoon of oil in a large cast iron or non-stick skillet over medium-high heat. Once hot, add the patties to the skillet and cook for about 4 minutes until the bottoms are nicely browned (you may need to do this in batches). Flip the patties and cook for another 3-4 minutes.
Reduce the heat to medium. Add 1/4 cup water to the skillet, cover with a lid, and let the patties cook for an additional 3-4 minutes.
While the patties are finishing, prepare the sauce. Combine all the sauce ingredients in a sauce pan over medium heat. Once the sauce begins to bubble, reduce the heat to low and simmer for 1-2 minutes, stirring occasionally, until the sauce thickens slightly.
Serve the hamburger steaks over steamed rice, and generously drizzle the sauce over the patties. Enjoy!