Cook the rice: Rinse the rice in cold water until the water runs mostly clear to remove excess starch. In a medium saucepan, combine the rice and 2 cups of water. Bring to a boil, then reduce the heat to low, cover tightly, and simmer for 15–18 minutes. Once done, remove from heat and let sit, covered, for 5 minutes. Fluff gently with a fork before serving.
Make the beef patties: In a bowl, mix the ground beef, finely chopped onion, Worcestershire sauce, soy sauce, salt, and pepper. Combine gently without overmixing to keep the patties tender. Form into four evenly sized patties, pressing a small indentation in the center of each to help them cook evenly.
Cook the patties: Heat a skillet over medium-high heat and lightly oil the surface. Cook the patties for 4–5 minutes per side, or until browned and cooked through. Remove the patties and set aside, reserving the drippings in the pan to flavor the gravy.
Make the brown gravy: Lower the heat to medium. Add the butter to the skillet with the drippings, then whisk in the flour to form a roux. Cook for about 1–2 minutes, stirring constantly, until golden and fragrant. Gradually whisk in the beef stock, scraping up any browned bits from the pan. Simmer for 5–7 minutes, stirring often, until the gravy thickens to a smooth consistency. Season with salt and pepper as needed.
