In a large mixing bowl whisk together the grated onion, crushed garlic, an egg, salt, pepper, and soy sauce. Add the ground beef and bread crumbs to the seasoning paste and mix with your hands, just until combined.
Divide and form the meat into 4 meatballs of equal size. Allow the meatballs to rest, covered in the refrigerator for 20 minutes to an hour.
After the patties have marinated a little while, preheat your cast iron skillet over medium heat. Meanwhile, flatten the meatballs into an 1/2" thick oval shape.
Press the patties into the hot skillet and sear the meat for 3-4 minutes on the first side. Flip the patties and sear for an additional 3-4 minutes on the other side. Remove the meat from the pan and set it onto a sheet pan. Transfer to a warm oven while you prepare the gravy.
Prepare the Brown Gravy
Pour the fat from the frying pan, reserving it in a heatsafe bowl. Wipe out any charred bits as they will cause your gravy to taste bitter. Pour 3 tablespoons of fat back into the pan and sprinkle the flour over it. Whisk until a paste forms. Cook the roux for 2-3 minutes on low heat until it begins bubbling and growing dark and begins to smell nutty.
Whisk the beef stock into the roux gradually to avoid lumps. Heat the gravy, whisking frequently, for 3-4 minutes or until the gravy thickens. Add the soy sauce, salt, and pepper, then give the gravy a taste and adjust the seasonings with more salt and pepper, if needed.
Once thickened, reduce the heat to warm and fry your egg. Be sure to stir the gravy occasionally to prevent a skin from forming on its surface. If the gravy isn't thick enough to coat a spoon, mix a quick slurry of 1 tablespoon of cornstarch and 1 tablespoon of cold water and whisk it into the gravy. Bring it to a simmer, stirring, until thickened.
Fry the Eggs (Optional)
Heat two tablespoons of oil in a skillet over medium-high heat.
Crack an egg (or two) into the oil and fry until the edges are crisp and the white is opaque- about 3 minutes. Repeat until your eggs are fried.
Spoon a bed of steamed white rice onto each plate. Place a beef patty on top of the rice and pour a generous amount of gravy over it. Top the patty with a sunny side-up egg and serve.
Leftovers can be stored in a food container and kept refrigerated for 3 days. Freezing:Freeze the beef mix prior to forming by wrapping in plastic wrap and freezing inside of a plastic storage bag. The raw mix will freeze for 3 months. Freeze the fried (cooked) beef patties in the gravy for 6 months. Thaw them under refrigeration and reheat until warmed through.