First, prepare the hamburger patties. In a large plate or a quarter baking sheet, line with a parchment paper. In a large bowl, combine ground beef, onion, panko, egg, mayonnaise, garlic salt and black pepper. Mix until just combined. Do not overmix. Divide into 6 portions, roll into balls, and pat down into thin round patties, about ⅓-inch thickness. Place on the prepare plate. If needed, use another parchment paper to separate the patties in between. Cover with plastic wrap and rest in the fridge for at least 30 minutes.
In the meantime, make the gravy. In a small saucepan over medium heat, melt the butter. Add flour and whisk continuously for 4 minutes. While whisking the flour roux, slowly pour in 1½ cups beef stock (not all the stock). Whisk until smooth. Start on cooking the mushroom topping now (see next step). Reduce heat to medium-low for the gravy, add soy sauce and browning seasoning sauce. Season with salt and pepper. Cook until the gravy is thick enough to coat the back of a spoon, whisking frequently. Set aside on stove top.
When the gravy is cooking, make the mushroom topping. In a medium skillet over high heat, warm ½ tablespoon vegetable oil, place mushrooms evenly on the skillet, then top with a layer of onion slices and sprinkle with garlic slices. Season with salt and pepper. Let the mushroom cook until caramelize, about 2 - 3 minutes. Reduce heat to medium. Stir and mix. Cook until the onions are softened, about 5 more minutes. Transfer to a medium bowl and set aside.
In a large skillet (preferably cast-iron) over high heat, warm ½ tablespoon vegetable oil. Season 3 hamburger patties with salt on both side. Cook until nicely charred and cooked through, about 2 - 3 minutes per side. Transfer to a paper towel lined plate. Wipe the skillet clean with paper towels. Add ½ tablespoon oil and repeat with the rest of the patties. Season and cook.
In another large skillet over medium heat, heat until hot. Melt 1 tablespoon butter. Add the eggs. When the outer edges turn white in color, cover with a lid. Turn down the heat to low and cook for 3 more minutes. Start on reheating the gravy (see next step).
When the eggs are cooking, work on the gravy. Reheat over medium-low heat, whisking for a couple times, about 3 minutes.
To serve, in serving plates, add steamed rice, hamburger patties (2 per person), mushroom topping, gravy and fried egg. Serve immediately.