Using a food processor fitted with a metal blade, process the mushrooms until very finely chopped.
In a large mixing bowl, combine the chopped mushrooms with ground beef, egg, panko, onion, soy sauce, Worcestershire, and pepper. Using your hands, gently mix together until well incorporated. Divide into quarters, and shape into four round patties.

In a large heavy-bottomed skillet set over medium-high heat, heat the canola oil. Add the patties, and cook for 4-5 minutes on each side, or until cooked all the way through on the inside and crisply browned on the outside. Transfer to a paper-towel lined plate, and tent loosely with aluminum foil to keep warm. Set aside.

Add the butter to the hot skillet. Once the butter starts to foam, add the sliced onions and saute for 7-10 minutes or until soft and golden. Stir in the flour and cook, stirring constantly, for about a minute. Whisk in the beef broth, and bring to a simmer, scraping up the crispy browned bits from the bottom of the pan.
Once the broth comes to a simmer, reduce heat to medium-low and continue cooking for 5-10 minutes, stirring frequently, until the gravy is thick and smooth.

While the gravy is simmering, pan-fry the eggs as you like them – I’m a fan of sunny side up, but over easy is just dandy, too.

To serve, scoop ½ cup cooked rice into a wide, shallow bowl. Top with a patty, and ladle a generous amount of gravy and onions overtop before finally topping off with an egg. Garnish with green onions, and dashes of sriracha and/or soy sauce.