Heat a large heavy sauté pan or wok over a medium-high heat. Add 1 tablespoon oil and swirl to coat the pan. When the oil is hot, add the onions, season with 1/2 teaspoon salt and 1/2 teaspoon pepper and cook until caramelized, 10-12 minutes. Set the onions aside. Reserve the pan.
Meanwhile, in a medium bowl, combine the beef, scallions, the garlic and 1 tablespoon of the soy sauce and season with pepper. Mix and form gently into 4 patties about ½ inch thick. Make an indentation with your thumb in the middle of each patty.
Heat the pan over a medium-high heat, add 1 tablespoon of the oil and swirl to coat the pan. Season the patties on both sides with salt and pepper, transfer to the pan and cook, turning once, for about 4 minutes for medium-rare, 5 minutes for well-done. Transfer the patties to a plate to rest for 5 minutes.
Return the pan to medium heat, and add the onions. Pour in the stock, bring to a simmer and add the remaining 1 teaspoon soy sauce. Whisk in the cornstarch slurry and simmer until the sauce is lightly thickened, about 1 minute . Taste to adjust the seasoning with salt and pepper, if necessary.
Break the eggs into a bowl. Heat a medium nonstick sauté pan over medium heat. Add the remaining 1 tablespoon of oil, and when hot, gently slide in the eggs.
Cook for about 1 minute, turn and cook for about 2 minutes until the whites are set and the yolks are still soft. (If you prefer your eggs sunny side up, add the eggs, cover, cook for about 1 minute, uncover, then cook until the whites are set and the yolks are still soft, 2 to 3 minutes). Season with salt and pepper.
Divide the rice among 4 soup plates or serving bowls, top with the burgers and top with the onions. With a nonmetal spatula, separate the eggs, transfer to the onions, garnish with the reserved scallion greens and serve.