In a large bowl, combine the ground meat, Worcestershire sauce, kosher salt, and black pepper. Mix gently and form into 4 equal patties. Set aside.
Cook the mushrooms
Heat 1 tablespoon olive oil in a medium pot over medium-high heat. Add the mushrooms and cook until browned and tender, about 7 minutes. Season with salt and black pepper. Transfer to a bowl and set aside.
Make the gravy
Reduce the heat to low and add the butter to the pot. Once melted, stir in the flour and cook, stirring often, until the roux turns peanut butter to light chocolate brown.
Add the onions and cook over low heat, stirring often, until very soft, about 8 minutes. Add the garlic and cook 2 more minutes.
Raise the heat to medium and slowly whisk in the beef broth. Add the Worcestershire sauce, soy sauce, and cooked mushrooms. Simmer until the gravy thickens enough to coat the back of a spoon. Keep warm on low.
Cook the patties
Lightly season the outside of the patties with a little more salt and black pepper.
Heat a large skillet over medium-high heat and add the cooking oil. Once the oil starts to lightly smoke, add the patties and sear until browned on the first side. Flip and brown the second side. Reduce the heat to medium-low and cook to your preferred doneness. Rest the patties while you cook the eggs.
Cook the eggs
Heat a nonstick skillet over medium heat and add 1 tablespoon olive oil. Crack in the eggs and cook until the whites are set and the yolks are still soft, or cook them to your liking.
Serve
To serve, spoon hot white rice onto each plate. Top with a patty, plenty of gravy, and a fried egg. Finish with scallions and crispy fried onions if using.
NOTES
Skip the Roux and use a Slurry: For a quicker gravy, skip the roux and use a slurry instead. Whisk 2 tablespoons cornstarch with 1/4 cup cold beef broth or water until smooth. Make the gravy as written, but leave out the butter and flour. Once the broth, soy sauce, Worcestershire, onions, garlic, and mushrooms come to a simmer, slowly whisk in the slurry and cook until thickened.I use a dark roux instead of a cornstarch slurry because it gives the gravy a deeper flavor and a richer color.Mushrooms are optional: Skip them and skip that cooking step if you want a simpler gravy.